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1
Zbl pre05768425
Doukhan, Paul (ed.); Lang, Gabriel (ed.); Surgailis, Donatas (ed.)
Dependence in probability and statistics.
(English)
[B] Lecture Notes in Statistics 200. Berlin: Springer. xv, 205~p. EUR~59.95/net; SFR~93.00; \sterling~53.99 (2010). ISBN 978-3-642-14103-4/pbk
MSC 2000:
*62-06 Proceedings of conferences (statistics)
65C50 Other computational problems in probability
2
Zbl 1189.60043
Doukhan, Paul; Neumann, Michael H.
The notion of $\psi$-weak dependence and its applications to bootstrapping time series.
(English)
[J] Probab. Surv. 5, 146-168, electronic only (2008). ISSN 1549-5787

Summary: We give an introduction to a notion of weak dependence which is more general than mixing and allows to treat for example processes driven by discrete innovations as they appear with time series bootstrap. As a typical example, we analyze autoregressive processes and their bootstrap analogues in detail and show how weak dependence can be easily derived from a contraction property of the process. Furthermore, we provide an overview of classes of processes possessing the property of weak dependence and describe important probabilistic results under such an assumption.
MSC 2000:
*60E15 Inequalities in probability theory
62E99 Statistical distribution theory

Keywords: autoregressive processes; autoregressive bootstrap; mixing; weak dependence

3
Zbl 1187.60013
Bardet, Jean-Marc; Doukhan, Paul; Lang, Gabriel; Ragache, Nicolas
Dependent Lindeberg central limit theorem and some applications.
(English)
[J] ESAIM, Probab. Stat. 12, 154-172 (2008). ISSN 1292-8100; ISSN 1262-3318

Summary: In this paper, a very useful lemma (in two versions) is proved: it simplifies notably the essential step to establish a Lindeberg central limit theorem for dependent processes. Then, applying this lemma to weakly dependent processes introduced in {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, No.~2, 313--342 (1999; Zbl 0996.60020)], a new central limit theorem is obtained for sample mean or kernel density estimator. Moreover, by using the subsampling, extensions under weaker assumptions of these central limit theorems are provided. All the usual causal or non causal time series: Gaussian, associated, linear, ARCH($\infty$), bilinear, Volterra processes, $\ldots$, enter this frame.
MSC 2000:
*60F05 Weak limit theorems
62G07 Curve estimation
62M10 Time series, etc. (statistics)
62G09 Statistical resampling methods

Keywords: central limit theorem; Lindeberg method; weak dependence; kernel density estimation; subsampling

Citations: Zbl 0996.60020

4
Zbl 1166.60031
Doukhan, Paul; Wintenberger, Olivier
Weakly dependent chains with infinite memory.
(English)
[J] Stochastic Processes Appl. 118, No. 11, 1997-2013 (2008). ISSN 0304-4149

The authors consider the stationary solution of the equation $X_t = F(X_{t-1},X_{t-2},\ldots;\xi_t)$ a.s. for $t\in \mathbb{Z}$ where $(\xi_t)_{t\in\mathbb{Z}}$ is a sequence of i.i.d. random variables, $F$ takes values in a Banach space and satisfies the Lipschitz-type condition. An explicit upper bound for the $\tau$-dependence coefficient introduced by {\it J. Dedecker} and {\it C. Prieur} [J. Theor. Probab. 17, No. 4, 855--885 (2004; Zbl 1067.60008)] of $(X_t)_{t\in\mathbb{Z}}$ is provided under specified assumptions concerning $F$ (involving certain Orlicz space). The relation between the above mentioned result and the analogous one for mixing coefficients established in [{\it M. Iosifescu, S. Grigorescu}, Dependence with Complete Connections and Applications. Cambridge Tracts in Mathematics. 96. (Cambridge), UK: Cambridge University Press. (1990; Zbl 0749.60067)] is discussed. SLLN, CLT and strong invariance principle are obtained using the bounds for $\tau(r)$ as $r\to \infty$. Among various examples one can mention non-linear autoregressive models and the Galton -- Watson process with immigration.
[Alexander V. Bulinski (Moskva)]
MSC 2000:
*60G99 Stochastic processes
62M10 Time series, etc. (statistics)
91B62 Dynamic economic models etc.
60K35 Interacting random processes
60F05 Weak limit theorems
60F17 Functional limit theorems

Keywords: weak dependence coefficient; SLLN; CLT; strong invariance principle.

Citations: Zbl 1067.60008; Zbl 0749.60067

5
Zbl 1148.60076
Doukhan, P.; Lang, G.; Louhichi, S.; Ycart, B.
A functional central limit theorem for interacting particle systems on transitive graphs.
(English)
[J] Markov Process. Relat. Fields 14, No. 1, 79-114 (2008). ISSN 1024-2953

Summary: A finite range interacting particle system on a transitive graph is considered. Assuming that the dynamics and the initial measure are invariant, the normalized empirical distribution process converges in distribution to a centered diffusion process. As an application, a central limit theorem for certain hitting times, interpreted as failure times of a coherent system in reliability, is derived.
MSC 2000:
*60K35 Interacting random processes
60F17 Functional limit theorems

Keywords: interacting particle system; functional central limit theorem; hitting time

6
Zbl pre05697515
Bardet, Jean-Marc; Doukhan, Paul; León, José Rafael
Uniform limit theorems for the integrated periodogram of weakly dependent time series and their applications to Whittle's estimate.
(English)
[J] J. Time Ser. Anal. 29, No. 5, 906-945 (2008). ISSN 0143-9782; ISSN 1467-9892

The authors prove the uniform convergence results such as a strong law of large numbers and a central limit theorem for the integrated periodogram of weakly dependent time series. The obtained results are applied to Whittle's approximation likelihood estimate. Numerous articles have been written on this estimation method after Whittle's seminal article. The asymptotic normality for Gaussian and causal linear, strong mixing and autoregressive conditionally heteroscedastic (ARCH)($\infty$) processes was established by {\it E. J. Hannan} [J. Appl. Probab. 10, 913 (1973; Zbl 0269.62075; Zbl 0261.62073)], {\it M. Rosenblatt} [Stationary Processes and Random Fields. Boston: Birkhaiiser Boston (1985)], {\it L. Giraitis} and {\it P. M. Robinson} [Econom. Theory 17, No. 3, 608--631 (2001; Zbl 1051.62074)]. The main goal of the present article is to provide a unified treatment of the asymptotic normality for very rich class of weakly dependent time processes, including those previously mentioned, as well as for some new classes of non-causal or nonlinear processes. Let $X = (X_k)_{k\in\mathbb Z}$ be a real-valued zero-mean fourth-order stationary time series, let $R(s)$ be the covariogram of $X$, and let $\kappa_4(i,j,k)$ be the fourth cumulants of $X$. \par The authors use Assumption M on $X$ which means that $\gamma=\sum_{s\in\mathbb Z}(R(s))^2<\infty$ and $\kappa_4=\sum_{i,j,k}\vert \kappa_4(i,j,k)\vert <\infty$. The periodogram of $X$ is $I_n(\lambda)=\frac{1}{2\pi n}\left\vert \sum_{k=1}^nX_ke^{-ik\lambda}\right\vert ^2$ for a $2\pi$-periodic function such that $g\in L^2(-\pi,\pi)$ define the integrated periodogram of $X$ and $J_n(g)=\int_{-\pi}^{\pi}g(\lambda)I_n(\lambda)d\lambda$ and $J(g)=\int_{-\pi}^{\pi}g(\lambda)f(\lambda)d\lambda$, where $f$ denoting the spectral density of $X$. The periodogram $I_n(\lambda)$ could be a natural but not consistent estimator of the spectral density. In the same time the behaviour of $J_n(g)$ becomes quite smooth, allowing an estimation of the spectral density. A special case of the integrated periodogram is Whittle's contrast, defined as a function $\beta\to J_n(h_{\beta})$, where $h_{\beta}$ is included in a class of functions depending on the vector of parameters ${\beta}$. The Whittle estimator minimizes this contrast. As a consequence, uniform limit theorems for the integrated periodogram $J_n(\cdot)$ are the appropriate tools for obtaining uniform limit theorems for Whittle's contrast, that imply limit theorems for Whittle's estimators. \par A uniform strong law of large numbers of integrated periodograms on a Sobolev-type space is first established only under Assumption M. Additional assumptions on the dependence properties of the time series have to be specified for establishing central limit theorems. The authors consider time series satisfying weak dependence properties introduced and developed in {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, No. 2, 313--342 (1999; Zbl 0996.60020)]. This frame of dependence includes a lot of models like causal or non-causal linear, bilinear, strong mixing processes, and dynamic systems. The presented results are obtained under weaker conditions on time series, but considering different functional spaces compared with those obtained in {\it R. Dahlhaus} [Stochastic Processes Appl. 30, No. 1, 69--83 (1988; Zbl 0655.60033)] or {\it T. Mikosch} and {\it R. Norvaiša} [Stochastic Processes Appl. 70, No. 1, 85--114 (1997; Zbl 0913.60032)]. Two frames of weak dependence are considered here. The first one exploits a causal property of dependence, the $\theta$-weak dependence property (see {\it Jérôme Dedecker} and {\it P. Doukhan} [Stochastic Processes Appl. 106, No. 1, 63--80 (2003; Zbl 1075.60513)]). Under certain conditions, the uniform limit theorems for integrated periodogram and asymptotic normality of Whittle's estimate are established. These general results are new and extend {\it E. J. Hannan} [J. Appl. Probab. 10, 913 (1973; Zbl 0269.62075)] and {\it M. Rosenblatt} [Stationary Processes and Random Fields. Boston: Birkhaiiser Boston (1985)] classical results for causal linear or strong mixing processes. The second type of dependence under consideration is $\eta$-weak dependence. This property allows us to derive central limit theorems for non-causal processes (see {\it P. Doukhan} and {\it O. Wintenberger} [Probab. Math. Stat. 27, No. 1, 45--73 (2007; Zbl 1124.60031)]). These results can be applied, for instance, to two-sided linear or Volterra processes.
[Mikhail P. Moklyachuk (Ky\"iv)]
MSC 2000:
*62M10 Time series, etc. (statistics)
60F17 Functional limit theorems
60F25 Lp-limit theorems (probability)
62M09 Non-Markovian processes: estimation
62M15 Spectral analysis of processes

Keywords: periodogram; weak dependence; Whittle estimate

7
Zbl 1165.62001
Dedecker, Jérôme; Doukhan, Paul; Lang, Gabriel; León, José Rafael R.; Louhichi, Sana; Prieur, Clémentine
Weak dependence. With examples and applications.
(English)
[B] Lecture Notes in Statistics 190. New York, NY: Springer. xiv, 318~p. EUR~46.95/net; SFR~82.00; \sterling~36.00; \$~59.95 (2007). ISBN 978-0-387-69951-6/pbk While for many years researchers were mostly working with independent samples, recent developments of methods for dependent data have moved the interest to such more complicated structures. There are numerous examples where dependent data occur. This leads to the problem of developing asymptotic results and limit theorems for sequences of dependent random variables. {\it P. Doukhan}, and {\it S. Louhichi} [A new weak dependence condition and applications to moment inequalities." Stochastic Processes Appl. 84, No. 2, 313--342 (1999; Zbl 0996.60020)] proposed a new concept of weak dependence which is more general than mixing and is suitable for almost all classes of processes of interest in statistics. Weak dependence refers to how a stochastic relationship between random variables decreases as the separation between them increases. The idea relates to measure asymptotic independence of a random process. This book continues this idea and provides a detailed description of the notion of weak dependence as well as properties and applications.\par As the authors say in the Preface, the book is organized in four parts: definitions and models, tools, limit theorems and applications. A brief introduction to the concept can be found in Chapter 1. Different classes of weakly dependent sequences are described in Chapter 2. Chapter 3 contains examples of random sequences with weak dependence, including Bernoulli shifts, Markov sequences and several time series processes. Chapters 4 and 5 are devoted to develop theory to base the forthcoming results in the later chapters and especially inequalities and moment bounds.\par Chapters 6 through 9 are devoted to developing limit theorems and asymptotic results, like strong laws of large numbers (Chapter 6), central limit theorems (Chapter 7), functional central limit theorems (Chapter 8), and laws of iterated logarithms (Chapter 9). Chapters 10 to 13 present interesting applications like empirical processes, functional and spectral estimation, and econometric applications respectively. The applications also refer to practical statistical problems like nonparametric statistics, spectral analysis, econometrics and resampling, offering robustness against deviations from standard independence assumptions. The book ends up with a detailed bibliography including some early attempts to the topic. \par Overall the book is neatly written, however, it is quite dense and hard for researchers without strong background on the topic. The exposition is too technical in several points. Despite its complexity notations are very consistent and this helps the readers considerably. On the other hand, the book is very rich in its material as it contains earlier works on dependence and tries to mention and show a lot of applications of the theory. It also contains a large number of examples and expositions of the idea of weak dependence in models like stable Markov chains, dynamical systems or more complicated models, nonlinear, non-Markovian, and heteroskedastic models with infinite memory which provide good insight. [Dimitris Karlis (Athens)] MSC 2000: *62-02 Research monographs (statistics) 60-02 Research monographs (probability theory) 60F05 Weak limit theorems 60G99 Stochastic processes 60F15 Strong limit theorems 60F17 Functional limit theorems 62H20 Statistical measures of associations Keywords: mixing conditions; limit theorems; asymptotic independence Citations: Zbl 0996.60020 8 Zbl 1144.60026 Doukhan, P.; Lang, G.; Surgailis, D. Randomly fractionally integrated processes. (English) [J] Lith. Math. J. 47, No. 1, 1-23 (2007); and Liet. Mat. Rink. 47, No. 1, 3-28 (2007). ISSN 0363-1672; ISSN 1573-8825 Authors' abstract: {\it A. Philippe, D. Surgailis}, and {\it M.-C. Viano} [C. R. Acad. Sci. Paris, Ser. 1. 342, 269--274 (2006; Zbl 1086.60506); in: Dependence in probability and statistics. New York, NY: Springer. Lect. Notes Statistics 187, 159--194 (2006; Zbl 05082993)] introduced two distinct time-varying mutually invertible fractionally integrated filters$A({\bold d}), B ({\bold d})$depending on an arbitrary sequence${\bold d} = (d_{t})_{t \in \Bbb Z}$of real numbers; if the parameter sequence is constant$d_{t} \equiv d$, then both filters$A({\bold d})$and$B ({\bold d})$reduce to the usual fractional integration operator$(1 - L)^{- d}$. They also studied partial sums limits of filtered white noise nonstationary processes$A({\bold d}) \epsilon_{t}$and$B ({\bold d}) \epsilon_{t}$for certain classes of deterministic sequences${\bold d}$. The present paper discusses the randomly fractionally integrated stationary processes$X_{t}^{A}= A ({\bold d}) \epsilon_{t}$and$X_{t}^{B}= B ({\bold d}) \epsilon_{t}$by assuming that${\bold d} = (d_{t}, t \in \Bbb Z)$is a random iid sequence, independent of the noise$(\epsilon_{t})$. In the case where the mean$\bar d = \mathbb{E}d_0 \in \left({0,1/2} \right)$, we show that large sample properties of$X^{A}$and$X^{B}$are similar to FARIMA$(0, \bar d, 0)$process; in particular, their partial sums converge to a fractional Brownian motion with parameter$\bar d + (1/2)$. The most technical part of the paper is the study and characterization of limit distributions of partial sums for nonlinear functions$h (X_{t}^{A})$of a randomly fractionally integrated process$X_{t}^{A}$with Gaussian noise. We prove that the limit distribution of those sums is determined by a conditional Hermite rank of$h$. For the special case of a constant deterministic sequence$d_{t}$, this reduces to the standard Hermite rank used by {\it R. L. Dobrushin} and {\it P. Major} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 50, No. 1, 27--52 (1979; Zbl 0397.60034)]. [Nikolai N. Leonenko (Cardiff)] MSC 2000: *60G10 Stationary processes 60F05 Weak limit theorems 60G18 Self-similar processes 60G12 General second order processes 60G15 Gaussian processes 60G35 Appl. of stochastic processes 62M20 Prediction, etc. (statistics) Keywords: fractional derivatives and integrals; self-similar processes; central limit theorem; noncentral limit theorems; FARIMA process Citations: Zbl 1086.60506; Zbl 0397.60034 9 Zbl 1140.60029 Doukhan, Paul; Truquet, Lionel A fixed point approach to model random fields. (English) [J] ALEA, Lat. Am. J. Probab. Math. Stat. 3, 111-132, electronic only (2007). ISSN 1980-0436 The authors prove the existence and uniqueness of the solution of the equation $$X_{t}=F((X_{t-j})_{j\in\mathbb{Z}^{d}\setminus\{0\}};\xi_{t}),$$ where the input$(\xi_{t})$is an i.i.d. random field. The proofs rely on the contraction principle hence Lipschitz type conditions are needed, but there are no assumptions relative to the conditional distribution. The solution writes$X_{t}=H((\xi_{t-s})_{s\in\mathbb{Z}^{d}}$and is a stationary random field with infinite interactions. The authors prove weak dependence properties of this solution. General results for stationary (non necessarily independent) input are also stated. Those results imply heavy restrictions on the innovations in some cases: a convenient notion of causality is thus used. [Mihai Gradinaru (Rennes)] MSC 2000: *60G60 Random fields 60B12 Limit theorems for vector-valued random variables (inf.-dim.case) 60F25 Lp-limit theorems (probability) 60K35 Interacting random processes 62M40 Statistics of random fields 60B99 Probability theory on general structures 60K99 Special processes Keywords: random fields; limit theorems for vector-valued random variables; interacting random processes; weak dependence; Bernoulli shifts 10 Zbl 1125.62100 Doukhan, Paul; Madre, Hélène; Rosenbaum, Mathieu Weak dependence for infinite ARCH-type bilinear models. (English) [J] Statistics 41, No. 1, 31-45 (2007). ISSN 0233-1888 Summary: {\it L. Giraitis} and {\it D. Surgailis} [ARCH-type bilinear models with double long memory. Stochastic Processes Appl. 100, No. 1--2, 275--300 (2002; Zbl 1057.62070)] introduced ARCH-type bilinear models for their specific long-range dependence properties. We rather consider weak-dependence properties of these models. The computation of mixing coefficients for such models does not look as an accessible objective. So, we resort to the notion of weak dependence introduced by {\it P. Doukhan} and {\it S. Louhichi} [A new weak dependence condition and applications to moment inequalities. ibid. 84, No. 2, 313--342 (1999; Zbl 0996.60020)], whose use seems more relevant here. The decay rate of the weak-dependence coefficients sequence is established under different specifications of the model coefficients. This implies various limit theorems and asymptotics for statistical procedures. We also derive bounds for the joint densities of this model in the case of regular inputs. MSC 2000: *62M10 Time series, etc. (statistics) 60F17 Functional limit theorems 60F05 Weak limit theorems Keywords: time series; ARCH models; GARCH models; weak dependence; Markov chain; Donsker invariance principle Citations: Zbl 1057.62070; Zbl 0996.60020 11 Zbl 1124.60031 Doukhan, Paul; Wintenberger, Olivier An invariance principle for weakly dependent stationary general models. (English) [J] Probab. Math. Stat. 27, No. 1, 45-73 (2007). ISSN 0208-4147 Summary: The aim of this paper is to refine a weak invariance principle for stationary sequences given by {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, No. 2, 313--342 (1999; Zbl 0996.60020)]. Since our conditions are not causal, our assumptions need to be stronger than the mixing and causal$\theta$-weak dependence assumptions used by {\it J. Dedecker} and {\it P. Doukhan} [Stochastic Processes Appl. 106, No. 1, 63--80 (2003; Zbl 1075.60513)]. Here, if moments of order greater than 2 exist, a weak invariance principle and convergence rates in the CLT are obtained; Doukhan and Louhichi [loc. cit.] assumed the existence of moments with order greater than 4. Besides the$\eta$- and$\kappa$-weak dependence conditions used previously, we introduce a weaker one,$\lambda$, which fits the Bernoulli shifts with dependent inputs. MSC 2000: *60F17 Functional limit theorems Keywords: weak dependence; Bernoulli shifts Citations: Zbl 0996.60020; Zbl 1075.60513 Cited in: Zbl pre05697515 12 Zbl 1117.60018 Doukhan, Paul; Neumann, Michael H. Probability and moment inequalities for sums of weakly dependent random variables, with applications. (English) [J] Stochastic Processes Appl. 117, No. 7, 878-903 (2007). ISSN 0304-4149 Summary: {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, 313--342 (1999; Zbl 0996.60020)] introduced a new concept of weak dependence which is more general than mixing. Such conditions are particularly well suited for deriving estimates for the cumulants of sums of random variables. We employ such cumulant estimates to derive inequalities of Bernstein and Rosenthal type which both improve on previous results. Furthermore, we consider several classes of processes and show that they fulfill appropriate weak dependence conditions. We also sketch applications of our inequalities in probability and statistics. MSC 2000: *60E15 Inequalities in probability theory 62E99 Statistical distribution theory Keywords: Bernstein inequality; cumulants; Rosenthal inequality; weak dependence Citations: Zbl 0996.60020 13 Zbl 1113.60038 Doukhan, Paul; Teyssière, Gilles; Winant, Pablo A$\text{LARCH}(\infty)$vector valued process. (English) [A] Bertail, Patrice (ed.) et al., Dependence in probability and statistics. New York, NY: Springer. Lecture Notes in Statistics 187, 245-258 (2006). ISBN 0-387-31741-4/pbk The purpose of this paper is to propose a unified framework for the study of$\text{ARCH}(\infty)$processes that are commonly used in the financial econometrics. The extension of the univariate$\text{ARCH}(\infty)$processes to the multidimensional case based in Volterra expansions is proposed. Existence and uniqueness in$L^p$sense is studied. Coupling, Markovian approximation and weak dependence conditions are discussed. [Nikolai N. Leonenko (Cardiff)] MSC 2000: *60G10 Stationary processes 60F05 Weak limit theorems 60G12 General second order processes 60G25 Prediction theory Keywords:$\text{ARCH}(\infty)$processes;$\text{LARCH}(\infty)$processes; vector processes; Volterra expansions; weak dependence 14 Zbl 1104.60017 Coupier, David; Doukhan, Paul; Ycart, Bernard Zero-one laws for binary random fields. (English) [J] ALEA, Lat. Am. J. Probab. Math. Stat. 2, 157-175, electronic only (2006). ISSN 1980-0436 Summary: A set of binary random variables indexed by a lattice torus is considered. Under a mixing hypothesis, the probability of any proposition belonging to the first-order logic of colored graphs tends to 0 or 1, as the size of the lattice tends to infinity. For the particular case of the Ising model with bounded pair potential and surface potential tending to$-\infty$, the threshold functions of local propositions are computed, and sufficient conditions for the zero-one law are given. MSC 2000: *60F20 Zero-one laws 60K35 Interacting random processes 15 Zbl 1096.62082 Doukhan, P.; Latour, A.; Oraichi, D. A simple integer-valued bilinear time series model. (English) [J] Adv. Appl. Probab. 38, No. 2, 559-578 (2006). ISSN 0001-8678 Summary: We extend the integer-valued model class to give a nonnegative integer-valued bilinear process, denoted by INBL$(p,q,m,n)$, similar to the real-valued bilinear model. We demonstrate the existence of this strictly stationary process and give an existence condition for it. The estimation problem is discussed in the context of a particular simple case. The method of moments is applied and the asymptotic joint distribution of the estimators is given: it turns out to be a normal distribution. We present numerical examples and applications of the model to real time series data on Meningitis and Escherichia coli infections. MSC 2000: *62M10 Time series, etc. (statistics) 60G10 Stationary processes 62E20 Asymptotic distribution theory in statistics 62F10 Point estimation Keywords: integer-valued process; bilinear model; central limit theorem 16 Zbl 1092.60002 Bertail, Patrice (ed.); Doukhan, Paul (ed.); Soulier, Philippe (ed.) Dependence in probability and statistics. (English) [B] Lecture Notes in Statistics 187. New York, NY: Springer. viii, 492~p. EUR~54.95/net; \sterling~42.50; \$~69.95 (2006). ISBN 0-387-31741-4/pbk

The articles of this volume will be reviewed individually.
MSC 2000:
*60-06 Proceedings of conferences (probability theory)
62-06 Proceedings of conferences (statistics)
00B15 Collections of articles of miscellaneous specific interest
17
Zbl 1073.60036
Doukhan, Paul; Lang, Gabriel; Surgailis, Donatas; Viano, Marie-Claude
Functional limit theorem for the empirical process of a class of Bernoulli shifts with long memory.
(English)
[J] J. Theor. Probab. 18, No. 1, 161-186 (2005). ISSN 0894-9840; ISSN 1572-9230

Let $X_t=Y_t+V_t,t\in Z=\{0,\pm1,\pm2,\dots\}$, be a strictly stationary process where $Y_t$ is a linear long memory process and $V_t$ is a nonlinear short memory process. More precisely, $Y_t=\sum_{i=0}^\infty b_i\zeta_{t-i}$ is a moving average process in independent and identically distributed random variables $\zeta_i,i\in Z$, with mean zero and variance one and with hyperbolically decaying coefficients $b_i\sim c_0i^{d-1}$ for some $d\in(0,1/2)$ and $c_0\not=0$. The short memory process $V_t$ is a Bernoulli shift $V_t=V(\zeta_t,\zeta_{t-1},\dots)$ where $V(z_0,z_1,\dots)$ is a Borel function defined on $R^{Z_+}$ with $Z_+=\{0,1,\dots\}$ which satisfies $$E^{1/2}[V(\zeta_0,\dots,\zeta_{-n},0,0,\dots)- V(\zeta_0,\dots,\zeta_{-n+1},0,0,\dots)]^2\leq Cn^{-\rho}$$ for some finite constant $C$ and $\rho>\max\{24-22d,13-11d+3(1-2d)/(4d)\}$. For every integer $N\geq1$, set $\overline{Y}_N=N^{-1}\sum_{t=1}^NY_t$ and $\widehat{F}_N(x)=N^{-1}\sum_{t=1}^NI(X_t\leq x),x\in R$. Let $F$ denote the marginal distribution function of $X_0$ and $f$ its desity. Under mild assumptions on the distribution of $\zeta_0$ it is shown that $\sup_{x\in R}N^{1/2-d}\vert \widehat{F}_N(x)-F(x)+f(x)\overline{Y}_N\vert =o_p(1)$. As a consequence, the processes $N^{1/2-d}(\widehat{F}_N-F)$ converge in distribution in the Skorokhod space $D[-\infty,\infty]$ to the degenerate process $\widetilde{c}fZ$, where $\widetilde{c}=(c_0^2B(d,2-2d)/ d(1+2d))^{1/2}$, with $B$ being the beta function, and where $Z$ is a standard normal random variable. To elucidate the role of the weakly dependent Bernoulli shift $V_t$, a discussion of several concrete examples is also included.
[Erich Häusler (Giessen)]
MSC 2000:
*60F17 Functional limit theorems
60G18 Self-similar processes
62M10 Time series, etc. (statistics)

Keywords: self-similar processes; time series

18
Zbl 1186.60080
Doukhan, Paul; León, José R.
Asymptotics for the $L^p$-deviation of the variance estimator under diffusion.
(English)
[J] ESAIM, Probab. Stat. 8, 132-149 (2004). ISSN 1292-8100; ISSN 1262-3318

Summary: We consider a diffusion process $X_t$ smoothed with (small) sampling parameter $\varepsilon$. As in {\it C. Berzin-Joseph, J.R. León} and {\it J. Ortega} [Stoch. Proc. Appl. 92, 11--30 (2001; Zbl 1047.60082)], we consider a kernel estimate $\widehat{\alpha }_{\varepsilon }$ with window $h(\varepsilon )$ of a function $\alpha$ of its variance. In order to exhibit global tests of hypothesis, we derive here central limit theorems for the $L^p$ deviations such as $$\frac{1}{\sqrt{h}}\left(\frac{h}{\varepsilon }\right)^{\frac{p}{2}}\left( \left\Vert \widehat{\alpha }_{\varepsilon }-{\alpha }\right\Vert _p^p- \Bbb{E}\left\Vert \widehat{\alpha }_{\varepsilon }-{\alpha }\right\Vert _p^p \right).$$
MSC 2000:
*60J65 Brownian motion
62M05 Markov processes: estimation
60F05 Weak limit theorems
60F25 Lp-limit theorems (probability)
60H05 Stochastic integrals
62M02 Markov processes: hypothesis testing

Keywords: Variance estimator; kernel; $L^p$-deviation; central limit theorem

Citations: Zbl 1047.60082

19
Zbl 1069.62070
Nze, Patrick Ango; Doukhan, Paul
Weak dependence: models and applications to econometrics.
(English)
[J] Econom. Theory 20, No. 6, 995-1045 (2004). ISSN 0266-4666; ISSN 1469-4360

Summary: We discuss weak dependence and mixing properties of some popular models. We also develop some of their econometric applications. Autoregressive models, autoregressive conditional heteroskedasticity (ARCH) models, and bilinear models are widely used in econometrics. More generally, stationary Markov modeling is often used. Bernoulli shifts also generate many useful stationary sequences, such as autoregressive moving average (ARMA) or ARCH$(\infty)$ processes. For Volterra processes, mixing properties are obtained given additional regularity assumptions on the distribution of the innovations. We recall associated probability limit theorems and investigate the nonparametric estimation of those sequences.
MSC 2000:
*62M10 Time series, etc. (statistics)
62P20 Appl. of statistics to economics

Keywords: unit root tests; bootstrap; generalized method of moments; Markovian models; functional estimation; empirical distributions

20
Zbl 1067.62085
Doukhan, Paul; Brandière, Odile
Dependent noise for stochastic algorithms.
(English)
[J] Probab. Math. Stat. 24, No. 2, 381-399 (2004). ISSN 0208-4147

Summary: We introduce different ways of being dependent for the input noise of stochastic algorithms. We are aimed to prove that such innovations allow to use the ODE (ordinary differential equation) method. Illustrations to the linear regression frame and to the law of large numbers for triangular arrays of weighted dependent random variables are also given.
MSC 2000:
*62L20 Stochastic approximation
62J05 Linear regression
21
Zbl 1075.60513
Dedecker, Jérôme; Doukhan, Paul
A new covariance inequality and applications.
(English)
[J] Stochastic Processes Appl. 106, No. 1, 63-80 (2003). ISSN 0304-4149

Three measures of dependence, i.e.\ strong mixing-type, mixingale-type and s-weak dependence coefficients are considered and compared and useful examples describing the behaviour of these coefficients are given. Further, a new covariance inequality based on mixingale type coefficient is proved and compared with a similar result developed recently for strong mixing sequences. New sufficient conditions to obtain sharp versions of Donsker invariance principle and Marcinkiewicz strong law of large numbers for weakly dependent sequences are established.
[Zuzana Prášková (Praha)]
MSC 2000:
*60F17 Functional limit theorems
60G10 Stationary processes
60G48 Generalizations of martingales

Keywords: strong mixing; mixingales; s-weak dependence; weak invariance principle; strong laws of large numbers

Cited in: Zbl pre05697515 Zbl 1124.60031

22
Zbl 1039.60013
Doukhan, Paul; Brandière, Odile
Algorithmes stochastiques à bruit dépendant (Dependent noise for stochastic algorithms).
(French)
[J] C. R., Math., Acad. Sci. Paris 337, No. 7, 473-476 (2003). ISSN 1631-073X

Summary: We introduce different ways of modeling the dependency of the input noise of stochastic algorithms. We are aimed to prove that such innovations allow us to use the ODE (ordinary differential equation) method. Illustrations in the linear regression framework and in the law of the large numbers for triangular arrays of weighted dependent random variables are also given. We have aimed to provide results easy to check in practice.
MSC 2000:
*60E15 Inequalities in probability theory

Keywords: modeling the dependency of the input noise; ordinary differential equation method; linear regression; law of the large numbers for triangular arrays

23
Zbl 1032.62081
Doukhan, Paul
Models, inequalities, and limit theorems for stationary sequences.
(English)
[A] Doukhan, Paul (ed.) et al., Theory and applications of long-range dependence. Boston, MA: Birkhäuser. 43-100 (2003). ISBN 0-8176-4168-8/hbk

Summary: Recently, the notion of dependence in time series has received major attention in the research literature. In statistics, the two types of dependence that are usually considered are weak and strong dependence. Even though these types of dependence correspond to real phenomena, they have not yet received a satisfactory definition. A major objective of statistics is to build consistent tests by using limit theorems in distribution. Thus, a natural perspective is to classify the statistical model by means of related limit theorems. We restrict our attention to two types of statistics: those based on partial sums and those based on Kolmogorov-Smirnov statistics.\par After listing some classes of stationary times series models, which are commonly used in statistics as well as in econometrics and finance, we shall recall some classes of weak dependence conditions and present the dependence properties of the previous models in terms of these dependence conditions.\par Limit theorems associated to Donsker and Kolmogorov statistics yield a first classification between weak and strong dependence situations, according to two kinds of considerations. Under weak dependence, the limit theorems have normalization of order $\sqrt n$, and the limiting processes are rather irregular (typically, Brownian motion), whereas strong dependence involves higher order normalizations and very regular limiting processes (such as products of deterministic functions and random variables). This classification is a first step towards a global time series analysis. A second step would be to generate statistics suitable to test the type of dependence in a stationary time series.
MSC 2000:
*62M10 Time series, etc. (statistics)
60F05 Weak limit theorems
60G18 Self-similar processes

Keywords: strong dependence; weak dependence; association; mixing; moment inequalities; empirical process; Kolmogorov-Smirnov; Donsker statistics; Markovian representation; Gaussian processes; linear processes; Bernoulli shifts

Cited in: Zbl 1126.62071

24
Zbl 1029.62035
Doukhan, Paul; Khezour, Abdelali; Lang, Gabriel
Nonparametric estimation for long-range dependent sequences.
(English)
[A] Doukhan, Paul (ed.) et al., Theory and applications of long-range dependence. Boston, MA: Birkhäuser. 303-311 (2003). ISBN 0-8176-4168-8/hbk

Summary: We consider density estimation and regression problems for long-range dependent processes. We recall here limit theorems for stationary Gaussian subordinated and infinite moving average processes. Let $f_n$ be a kernel functional estimate defined from a sequence of $n$ observations of the process. We observe strong differences in the convergence of the estimate compared to the case of weak dependent processes. Denoting $\varphi_n(x)$ the recentered and rescaled estimate $\text{(Var} f_n(x))^{-1/2} [f_n(x)-f(x)]$, we have:\par $\bullet$ $\varphi_n(x) @> d >> \sigma(x){\cal Z}$, for long-range dependent processes, where $Z$ is a random variable independent of $x$ so that the limiting process is degenerated.\par $\bullet$ $\varphi_n(x) @>fidi>> \dot W(x)$ for weak dependent processes, where $\dot W$ is a Gaussian process.\par This difference may be a first step to a test of the dependence type of a process.
MSC 2000:
*62G07 Curve estimation
62M10 Time series, etc. (statistics)
62G08 Nonparametric regression
62G20 Nonparametric asymptotic efficiency

Keywords: long-range dependence; kernel density estimator; kernel regression estimator; noncentral limit theorem

25
Zbl 1005.00017
Doukhan, Paul (ed.); Oppenheim, George (ed.); Taqqu, Murad S. (ed.)
Theory and applications of long-range dependence.
(English)
[B] Boston, MA: Birkhäuser. x, 716 p. EUR 128.00/net; sFr. 198.00 (2003). ISBN 0-8176-4168-8/hbk

The articles of this volume will be reviewed individually.
MSC 2000:
*00B15 Collections of articles of miscellaneous specific interest
60-06 Proceedings of conferences (probability theory)
62-06 Proceedings of conferences (statistics)

Keywords: Long-range dependence

26
Zbl 1061.60016
Doukhan, Paul; Lang, Gabriel
Rates in the empirical central limit theorem for stationary weakly dependent random fields.
(English)
[J] Stat. Inference Stoch. Process. 5, No. 2, 199-228 (2002). ISSN 1387-0874; ISSN 1572-9311

The authors derives weak dependence conditions as the natural generalization to random fields on notions developed by {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, 313--342 (1999; Zbl 0996.60020)]. Examples of such weakly dependent fields are also defined. In the context of a weak dependence coefficient series with arithmetic or geometric decay, the authors give explicit bounds in the Prokhorov metric for the convergence in the empirical central limit theorem. For random fields indexed by ${\Bbb Z}^d ,$ in the geometric decay case, rates have the form $n^{-1/(8d+24)}L(n)$, where $L(n)$ is a power of $\log (n).$
[Zdzisław Rychlik (Lublin)]
MSC 2000:
*60F05 Weak limit theorems
60G60 Random fields
60F17 Functional limit theorems
60G10 Stationary processes

Keywords: stationary sequences; inequalities; Rosenthal inequality; positive dependence; mixing; central limit theorem

Citations: Zbl 0996.60020

27
Zbl 1030.60016
Ango Nze, Patrick; Doukhan, Paul
Weak dependence: Models and applications.
(English)
[A] Dehling, Herold (ed.) et al., Empirical process techniques for dependent data. Boston, MA: Birkhäuser. 117-136 (2002). ISBN 0-8176-4201-3/hbk

This paper aims at a systematic introduction to a new weak dependence condition. The new dependence condition gives access to some classes of models which could not be studied by standard tools such as mixing. The authors also show that some standard models satisfy this property, including stationary Markov models, bilinear models, and more generally, Bernoulli shifts. In addition, the new dependence condition is applied to derive a weak Donsker invariance principle and the empirical CLT.
[Zdzisław Rychlik (Lublin)]
MSC 2000:
*60F05 Weak limit theorems
62G07 Curve estimation

Keywords: weak dependence; Donsker invariance principle; empirical central limit theorem

28
Zbl 1016.60059
Doukhan, Paul; Lang, Gabriel; Surgailis, Donatas
Asymptotics of weighted empirical processes of linear fields with long-range dependence.
(English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 38, No.6, 879-896 (2002). ISSN 0246-0203

Authors' summary: We discuss the asymptotic behavior of weighted empirical processes of stationary linear random fields in $\bbfZ^d$ with long-range dependence. It is shown that an appropriately standardized empirical process converges weakly in the uniform topology to a degenerated process of the form $fZ$, where $Z$ is a standard normal random variable and $f$ is the marginal probability density of the underlying random field.
[B.L.S.Prakasa Rao (New Delhi)]
MSC 2000:
*60G60 Random fields
60F17 Functional limit theorems

Keywords: linear random fields; long-range dependence

29
Zbl 1012.62037
Ango Nze, Patrick; Bühlmann, Peter; Doukhan, Paul
Weak dependence beyond mixing and asymptotics for nonparametric regression.
(English)
[J] Ann. Stat. 30, No.2, 397-430 (2002). ISSN 0090-5364

Summary: We consider a new concept of weak dependence, introduced by {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, No. 2, 313-342 (1999; Zbl 0996.60020)], which is more general than the classical frameworks of mixing or associated sequences. The new notion is broad enough to include many interesting examples such as very general Bernoulli shifts, Markovian models or time series bootstrap processes with discrete innovations. \par Under such a weak dependence assumption, we investigate nonparametric regression which represents one (among many) important statistical estimation problem. We justify in this more general setting the whitening by windowing principle'' for nonparametric regression, saying that asymptotic properties remain in first order the same as for independent samples. The proofs borrow from previously used strategies, but precise arguments are developed under the new aspect of general weak dependence.
MSC 2000:
*62G08 Nonparametric regression
62M10 Time series, etc. (statistics)
60F05 Weak limit theorems

Citations: Zbl 0996.60020

30
Zbl 0973.62030
Doukhan, Paul; Louhichi, Sana
Functional estimation of a density under a new weak dependence condition.
(English)
[J] Scand. J. Stat. 28, No.2, 325-341 (2001). ISSN 0303-6898; ISSN 1467-9469

The authors analyze properties of usual kernel density estimators in the case where the data $X_1$,\dots,$X_n$,\dots is a time series satisfying a new weak dependence condition:\par for any bounded $f:R^n\to R$,\quad $g: R^m\to R$,\quad $i_1\le\dots\le i_n<i_n+r\le j_1\le\dots\le j_m$, $$|\text{Cov}(f(X_{i_1},\dots,X_{i_n}),g(X_{j_1},\dots,X_{j_m}))|\le C \text{Lip}(f)\text{Lip}(g)\vartheta_r$$ or $$|{\text Cov}(f(X_{i_1},\dots,X_{i_n}),g(X_{j_1},\dots,X_{j_m}))|\le C\min\{\text{Lip}(f),\text{Lip}(g)\}\vartheta_r,$$ where $\text {Lip}$ is the Lipschitz modulus and $\vartheta_r$ is some fixed number sequence $\vartheta_r\to 0$ as $r\to\infty$. It is demonstrated that such inequalities can be derived with various $\vartheta_r$ for Bernoulli shift sequences, e.g., for Volterra processes and ARMA and bilinear processes. The authors derive bias and MISE asymptotics, asymptotic normality results and a.s. convergence properties for kernel estimates under these mixing conditions. E.g., if $\vartheta_r=O(r^{-12-\nu})$ then asymptotic normality holds true.
[R.E.Maiboroda (Ky\" iv)]
MSC 2000:
*62G07 Curve estimation
62M10 Time series, etc. (statistics)
62G20 Nonparametric asymptotic efficiency

Keywords: central limit theorem; inequalities; mixing; positive dependence; Rosenthal inequality; stationary sequences; kernel density estimators

31
Zbl 0956.60006
Coulon-Prieur, Clémentine; Doukhan, Paul
A triangular central limit theorem under a new weak dependence condition.
(English)
[J] Stat. Probab. Lett. 47, No.1, 61-68 (2000). ISSN 0167-7152

The central limit theorem is proved for triangular arrays under a new weak dependence condition which is a variation of that from {\it P. Doukhan} and {\it S. Louhichi} [Stochastic Processes Appl. 84, 313-342 (1999)]. The definition of such a weak dependence extends on strong mixing and includes non-mixing Markov processes and associated or Gaussian sequences. The theorems proved in this paper apply for linear arrays and standard kernel density estimates under weak dependence and lead to an extension of results of {\it M. Peligrad} and {\it S. Utev} [Ann. Probab. 25, No. 1, 443-456 (1997; Zbl 0876.60013)]. The method of proof is a variation of Lindeberg method after {\it E. Rio} [ESAIM, Probab. Stat. 1, 35-61 (1997; Zbl 0869.60021) and Probab. Theory Relat. Fields 104, No. 2, 255-282 (1996; Zbl 0838.60017)].
[Birute Kryžien\D{e} (Vilnius)]
MSC 2000:
*60F05 Weak limit theorems
60F17 Functional limit theorems
60G10 Stationary processes
60G99 Stochastic processes
60E15 Inequalities in probability theory

Keywords: stationary sequences; Lindeberg theorem; central limit theorem; non-parametric estimation; $s$- and $w$-weakly dependence

Citations: Zbl 0876.60013; Zbl 0869.60021; Zbl 0838.60017

Cited in: Zbl 1067.60008

32
Zbl 0996.60020
Doukhan, P.; Louhichi, S.
A new weak dependence condition and applications to moment inequalities.
(English)
[J] Stochastic Processes Appl. 84, No.2, 313-342 (1999). ISSN 0304-4149

A new weak dependence condition for random sequences is proposed which is formulated in terms of covariances between past and future observations. It is proved that the new definition includes mixing sequences, functions of associated and Gaussian sequences as well as Bernoulli shifts and models with Markovian representation. A version of functional central limit theorem under the considered type of dependence is proved and an invariance principle for empirical processes is established.
[Zuzana Prášková (Praha)]
MSC 2000:
*60E15 Inequalities in probability theory
60F17 Functional limit theorems
60G10 Stationary processes
60F05 Weak limit theorems

Keywords: stationary sequences; weak dependence; mixing sequences; association; Rosenthal inequality; Marcinkiewicz-Zygmund inequality

33
Zbl 0939.60006
Hariz, Samir Ben; Doukhan, Paul; Léon, José Rafael
Central limit theorem for the local time of a Gaussian process.
(English)
[A] Dalang, Robert C. (ed.) et al., Seminar on Stochastic analysis, random fields and applications. Centro Stefano Franscini, Ascona, Italy, September 1996. Basel: Birkhäuser. Prog. Probab. 45, 25-37 (1999). ISBN 3-7643-6106-9

Let $\{X_t,\ t\in R\}$ be a real-valued, Gaussian, stationary process with $E(X_s)=0$, $E(X^2_s)=1$ and covariance function $E(X_s,X_t)=r(|s-t|)$. As is known, under the first of the conditions $$1.\ \int^\infty_0 {ds \over \bigl(1- r^2(s)\bigr)^p}< \infty, \quad 2.\ \exists m>0 : \int\bigl |r(s)\bigr|^m ds< \infty,$$ with $p={1\over 2}$ the so-called local time $l_t(x)$ exists and admits the following expansion in $L^2(\Omega)$: $$l_t (x)=p(x)\sum^\infty_{k=0}{H_k(x)\over k!}\int^t_0H_k(X_s)ds,$$ where $p(x)={1 \over \sqrt{2\pi}}e^{-x^2/2}$, $H_k(x)=(-1)^k{p^{(k)}(x)\over p(x)}$ denote the normal density and the $k$th order Hermite polynomial, respectively. \par It is proved that under conditions (1) the remainder of this expansion, $$R_t (x)=p(x) \sum^\infty_{k=m} {H_k(x)\over k!}\int^t_0H_k(X_s)ds,$$ suitably normalized, is asymptotically Gaussian for some $m$, chosen as the smallest number, satisfying (1.2). Moreover, it is shown that under these conditions: 1) with $p={1\over 2}$ the finite-dimensional distributions of the random process $\{{1\over\sqrt t}R_t (x),\ x\in R\}$ converge to those of a Gaussian and centered process $R(x)$ with covariance function $$\Gamma(x,y)=2 \sum^\infty_{k=m} {H_k(x)p(x) H_k(y)p(y) \over k!}\int^\infty_0r^k(s)ds,$$ 2) with $p>1/2$ the limiting process $R(\cdot)$ admits a modification in law with almost surely continuous sample paths, and 3) for $p>1$ the previous convergence is functional (in the space $C(R,R))$.
[S.V.Zhulenev (Moskva)]
MSC 2000:
*60F05 Weak limit theorems
60G15 Gaussian processes

Keywords: local time of real stationary Gaussian process; central limit theorem

34
Zbl 0951.62074
Ango Nze, P.; Doukhan, P.
Functional estimation for time series: Uniform convergence properties.
(English)
[J] J. Stat. Plann. Inference 68, No.1, 5-29 (1998). ISSN 0378-3758

The authors deal with the estimation of the density of the marginal distribution of $X_1$ and of the regression function $r(x)= E(Y_1\mid X_1=x)$ relative to $Z$ for a strongly mixing stationary process $Z= (X_n, Y_n)_{n\in N^*}$. For this purpose they extend the results of {\it G. Walter} and {\it J. Blum} [Ann. Stat. 7, 328-340 (1979; Zbl 0403.62025)] on probability density estimation using delta sequences. They show that variance bounds for the estimates achieve minimax convergence rates.
[T.Cipra (Praha)]
MSC 2000:
*62M10 Time series, etc. (statistics)
62G08 Nonparametric regression
62G05 Nonparametric estimation
62J02 General nonlinear regression
60G10 Stationary processes

Keywords: autoregressive processes; mixing

Citations: Zbl 0403.62025

35
Zbl 0948.60012
Doukhan, Paul; Surgailis, Donatas
Théorème de limite centrale fonctionnelle pour la fonction de répartition empirique d'un processus linéaire à mémoire courte. (Functional central limit theorem for the empirical process of short memory linear processes.)
(English. Abridged French version)
[J] C. R. Acad. Sci., Paris, Sér. I, Math. 326, No.1, 87-92 (1998). ISSN 0764-4442

Consider a strictly stationary causal linear sequence $\{X_j,\ j\in {\Bbb Z}\}$, where $X_j=\sum_{t\ge 0}a_t\xi_{j-t}$, for $j\in {\Bbb Z}$, $\{\xi_j,\ j\in{\Bbb Z}\}$ is an independently and identically distributed sequence and $a_j,\ j\ge 0$, are (nonrandom) weights such that $\sum_{t\ge 0}|a_t|^\gamma<\infty$ and $E|\xi_0|^{4\gamma}<\infty$ for some $\gamma\in(0,\infty]$. Assume that $v\text{ var}(X_j)<\infty$ and the process becomes a short memory linear process. The authors prove the functional central limit theorem for the empirical distribution function of this process under the condition that $|E\exp{iui_0}|\le C/(1+|u|)^{\Delta}$, for any $u\in {\Bbb R}$, some $C<\infty$, and $1/2<\Delta\le 1$.
[Y.Wu (North York)]
MSC 2000:
*60F05 Weak limit theorems

Keywords: central limit theorem; short memory linear process; empirical distribution

36
Zbl 0927.60007
Doukhan, Paul
An overview on weak dependence of stationary sequences.
(English)
[J] Acta Cient. Venez. 49, No.2, 78-93 (1998). ISSN 0001-5504

This is an expository paper about recent topics on strong and weak dependence of stationary sequences and related limit theorems. It overviews, among others, the relations between some stochastic sequences and weak dependence conditions. For example, it treats Gaussian processes satisfying some mixing condition, linear processes satisfying $L$-weak dependence condition and associated processes.
[K.-i.Yoshihara (Tokyo)]
MSC 2000:
*60-02 Research monographs (probability theory)
60F05 Weak limit theorems
62G05 Nonparametric estimation

Keywords: strong dependence; weak dependence; empirical process; central limit theorem; functional estimation

37
Zbl 0919.62029
Doukhan, Paul; Louhichi, Sana
Functional estimation of a density under a weak dependence condition. (Estimation de la densité d'une suite faiblement dépendante.)
(French)
[J] C. R. Acad. Sci., Paris, Sér. I, Math. 327, No.12, 989-992 (1998). ISSN 0764-4442

Summary: The purpose of this paper is to prove through the analysis of the behaviour of a standard kernel density estimator that the notion of weak dependence defined in a previous paper [the first author, Prépubl. 97.08, Univ. Paris-Sud (1997)] has sharp properties enough to be used in various situations. This weak dependence condition extends the previously defined ones such as mixing, association, and it allows to consider new classes such as weak shift processes based on independent sequences as well as some non-mixing Markov processes.
MSC 2000:
*62G07 Curve estimation
62M99 Inference from stochastic processes

Keywords: kernel density estimator; weak dependence

38
Zbl 0893.62024
Ango Nze, P.; Doukhan, P.
Functional estimation for time series. I: Quadratic convergence properties.
(English)
[J] Math. Methods Stat. 5, No.4, 404-423 (1996). ISSN 1066-5307; ISSN 1934-8045

Summary: Let ${\bold Z}= (X_n,Y_n)_{n\in\bbfN^*}$ be a strongly mixing stationary stochastic process. We consider delta-estimates of the density of the marginal distribution of $X_1$ and of the regression function $r(x)= \bbfE[Y_1\mid X_1=x]$ for a class of estimators proposed formerly by {\it G. Walter} and {\it J. Blum} [Ann. Stat. 7, 328-340 (1979; Zbl 0403.62025)]. A finer evaluation of the variance of these estimates may be undertaken thanks to a new covariance inequality. The bounds reach an optimal order (that is of the i.i.d.'s). Optimal bounds for MISE criterion are deduced from this basic result. We also deduce the convergence in law of some quadratic functionals. We examine assumptions of strong dependence and of absolute regularity. Minimax rates are established.
MSC 2000:
*62G07 Curve estimation
62M10 Time series, etc. (statistics)
62J02 General nonlinear regression
60G10 Stationary processes
60G99 Stochastic processes
62G20 Nonparametric asymptotic efficiency

Keywords: autoregressive process; delta-estimates

Citations: Zbl 0403.62025

39
Zbl 0881.60023
Doukhan, Paul; León, Jose R.; Soulier, Philippe
Central and non central limit theorems for quadratic forms of a strongly dependent Gaussian field.
(English)
[J] REBRAPE 10, No.2, 205-223 (1996). ISSN 0103-0752

Summary: Strong dependence for a random field means that the autocorrelation function is not summable. In this case the usual central limit theorem for quadratic forms of the field does not necessarily hold. {\it R. Fox} and {\it M. S. Taqqu} [Ann. Probab. 13, 428-446 (1985; Zbl 0569.60016) and Probab. Theory Relat. Fields 74, 213-240 (1987; Zbl 0586.60019)] have studied the case of Gaussian processes. Using a representation of the quadratic form as a multiple Itô-Wiener stochastic integral, we extend these results to the case of Gaussian fields. In the one-dimensional case, our non central theorem is an extension of the result of Fox and Taqqu (1985).
MSC 2000:
*60F05 Weak limit theorems
60G60 Random fields

Keywords: Gaussian fields; multiple stochastic integrals; strong dependence; multiple Itô-Wiener stochastic integral; noncentral theorem

Citations: Zbl 0569.60016; Zbl 0586.60019

40
Zbl 0874.60046
Doukhan, P.; León, J.R.
Asymptotics for the local time of a strongly dependent vector-valued Gaussian random field.
(English)
[J] Acta Math. Hung. 70, No.4, 329-351 (1996). ISSN 0236-5294; ISSN 1588-2632

The authors study the local time of normalized vector-valued Gaussian random fields $X=(X_t; t\in\bbfR^d)$, $X_t\in\bbfR^p$. Extending a result of {\it S. M. Berman} [Stochastic Processes Appl. 12, 1-26 (1981; Zbl 0471.60082)] they obtain a series expansion of the local time $\ell_X$ in terms of multidimensional Hermite polynomials. Moreover, for $X$ stationary with a covariance matrix of long range dependence type, the authors apply a theorem of {\it M. V. Sanchez de Naranjo} [J. Multivariate Anal. 44, No. 2, 227-255 (1993; Zbl 0770.60025)] to deduce, from this series expansion, the asymptotic behaviour of the local time $\ell_X([0,\tau]^d,x)$ $(x\in\bbfR^p)$ as $\tau\to\infty$, in terms of random spectral measures associated to $X$. Finally, the authors consider a sufficiently regular bijective transform $Y$ of $X$ ($X$ stationary) and give series expansions of the kernel estimates for the marginal density of $Y$. This extends a result of {\it M. Rosenblatt} [NSF-CBMS Regional Conference Series in Probability and Statistics 3 (1991)].
[M.Dozzi (Nancy)]
MSC 2000:
*60G60 Random fields
60F05 Weak limit theorems

Keywords: local time; asymptotic behaviour; Edgeworth expansion; long range dependence; Gaussian random fields; Hermite polynomials; stationary; random spectral measures

Citations: Zbl 0471.60082; Zbl 0770.60025

41
Zbl 0861.60005
Doukhan, P.; Gamboa, F.
Superresolution rates in Prokhorov metric.
(English)
[J] Can. J. Math. 48, No.2, 316-329 (1996). ISSN 0008-414X; ISSN 1496-4279

Summary: Consider the problem of recovering a probability measure supported by a compact subset $U$ of $\bbfR^m$ when the available measurements concern only some of its $\Phi$-moments ($\Phi$ being an $\bbfR^k$ valued continuous function on $U$). When the true $\Phi$-moment $c$ lies on the boundary of the convex hull of $\Phi(U)$, generalizing the results of the second author and {\it E. Gassiat} [SIAM J. Math. Anal. 27, No. 4, 1129-1152 (1996)], we construct a small set $R_{\alpha,\delta(\varepsilon)}$ such that any probability measure $\mu$ satisfying $|\int_U \Phi(x)d\mu(x)-c|\le\varepsilon$ is almost concentrated on $R_{\alpha,\delta(\varepsilon)}$. When $\Phi$ is a pointwise $T$-system (extension of $T$-systems), the study of the set $R_{\alpha,\delta(\varepsilon)}$ leads to the evaluation of the Prokhorov radius of the set $\{\mu: |\int_U \Phi(x)d\mu(x)-c|\le \varepsilon\}$.
MSC 2000:
*60A10 Probabilistic measure theory
43A07 Means on groups, etc.
62A01 Foundational and philosophical topics

Keywords: probability measure; Prokhorov radius

42
Zbl 0817.60028
Doukhan, Paul; Massart, Pascal; Rio, Emmanuel
Invariance principles for absolutely regular empirical processes.
(English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 31, No.2, 393-427 (1995). ISSN 0246-0203

Let $(\xi\sb i)\sb{i \in \bbfZ}$ be a strictly stationary sequence of random elements of Polish space $X$ with common distribution $P$. Let ${\germ F}\sb 0 = \sigma (\xi\sb i : i \le 0)$ and ${\germ G}\sb n = \sigma (\xi\sb i : i \ge n)$ and define the absolutely regular mixing coefficient $$\beta\sb n = {1 \over 2} \sup \sum\sb{(i,j) \in I \times J} \bigl \vert P(A\sb i \cap B\sb j) - P(A\sb i) P(B\sb j) \bigr \vert$$ where the supremum is taken over all the finite partitions $(A\sb i)\sb{i \in I}$ and $(B\sb j)\sb{j \in J}$ of ${\germ F}\sb 0$ and ${\germ G}\sb n$, respectively.\par Suppose $(\xi\sb i)\sb{i \in \bbfZ}$ is the absolutely regular sequence with mixing coefficient $\beta\sb n$ satisfying the summability condition $\sum\sp \infty\sb{n=1} \beta\sb n < \infty$. Define the mixing rate function $\beta (t) = \beta\sb{[t]}$ $(t \ge 1)$ and $\beta (t) = 1$ otherwise. For any numerical function $f$ define a new norm for $f$ by $$\Vert f \Vert\sb{2, \beta} = \left[ \int\sp 1\sb 0 \beta\sp{-1} (u) \bigl[ Q\sb f(u) \bigr]\sp 2du \right]\sp{1/2}$$ where $\beta\sp{-1}$ denotes the càdlàg inverse of the monotonic function $\beta (\cdot)$ and $Q\sb f$ denotes the quantile function of $\vert f(\xi\sb 0) \vert$. Let ${\cal F}$ be a class of numerical functions with $\Vert f \Vert\sb{2, \beta} < \infty$. Let $P\sb n$ be the empirical probability measure $P\sb n = \sum\sp n\sb{i=1} \delta\sb{\xi\sb i}$ and define $Z\sb n = \sqrt n(P\sb n - P)$. The authors prove that a functional invariance principle in the sense of Donsker holds for $\{Z\sb n(f) : f \in {\cal F}\}$ under some conditions. The results extend the previous result of the authors [ibid. 30, No. 1, 63-82 (1994; Zbl 0790.60037)].
[K.-i.Yoshihara (Yokohama)]
MSC 2000:
*60F17 Functional limit theorems
62G99 Nonparametric inference

Keywords: absolutely regular; empirical measure; quantile function; entropy with bracketing; functional invariance principle; strictly stationary sequence of random elements

Citations: Zbl 0790.60037

Cited in: Zbl 0836.60028 Zbl 0874.62052

43
Zbl 0803.60003
Doukhan, Paul; Gamboa, Fabrice
Superresolution rates in Prokhorov metric. (Vitesses de superrésolution en distance de Prokhorov.)
(French. Abridged English version)
[J] C. R. Acad. Sci., Paris, Sér. I 318, No.12, 1143-1148 (1994). ISSN 0764-4442

Summary: Consider the problem of recovering a probability measure supposed by the compact set $[0,1]$, when the available measurements concern only some of its $\Phi$-moments $(\Phi$ being an $\bbfR\sp k$ valued continuous function on $[0,1])$. When the true $\Phi$-moment $c$ lies on the boundary of the convex hull of $\Phi([0, 1])$, generalizing the results of the second author and {\it E. Gassiat} [Sets of superresolution and the maximum entropy method on the mean (not yet published)], we construct a small Lebesgue measure set $R\sb{\alpha, \delta (\varepsilon)}$ such that any probability measure $\mu$ satisfying $$\biggl\Vert \int\sb{[0,1]} \Phi(x) d\mu(x) -c\biggr\Vert \leq\varepsilon$$ is almost concentrated on $R\sb{\alpha, \delta (\varepsilon)}$. When $\Phi$ is assumed to be a punctual $T$-system, the description of $R\sb{\alpha, \delta (\varepsilon)}$ allows the evaluation of the Prokhorov radius of the set $$\biggl\{ \mu: \biggl\Vert \int\sb{[0,1]} \Phi(x) d\mu(x) -c \biggr\Vert \leq \varepsilon \biggr\}.$$
MSC 2000:
*60B05 Probability measures on topological spaces

Keywords: weakly determinate points; pseudo-Haar regularity condition; superconcentration properties; punctual $T$-system; Prokhorov radius

44
Zbl 0801.60027
Doukhan, Paul
Mixing: Properties and examples.
(English)
[B] Lecture Notes in Statistics (Springer). 85. New York: Springer-Verlag. xii, 142 p. DM 68.00; öS 530.40; sFr. 68.00 (1994). ISBN 0-387-94214-9

The components of some random processes or fields are in a certain sense weakly dependent. One of the different ways to measure this kind of dependence is introducing the so-called mixing coefficients. In the first half of the present monograph the general properties of these coefficients are investigated. The coefficients in question are the strong mixing $(\alpha$-mixing), the uniform mixing $(\varphi$-mixing), the $*$-mixing $(\psi$-mixing) and the maximal correlation $(\rho$- mixing) coefficients and the coefficient of absolute regularity $(\beta$- mixing coef.). The author gives a short description of the relations between these various coefficients and discusses the difference of mixing for processes and for fields. Then he develops the tools now available for working with mixing processes and fields. These are covariance inequalities, Berbee's and Bradley's reconstruction theorems, a Rosenthal-type moment inequality, variations of Hoeffding's and Bernstein's exponential inequalities and maximal inequalities. This part of the book is finished by quotation of some results concerning the central limit theorem (for instance dimension dependent convergence rates).\par Examples for mixing processes and fields are considered in the second half of the monograph. For discrete Gaussian fields, Gibbs fields, linear fields and Markov chains with general state space the author gives conditions implying convergence of the mixing coefficients at fixed (e.g. geometric) rates. Various models of auto-regressive sequences are treated as special cases of Markov chains. For time-continuous Markov processes, mixing properties are deduced by considering the infinitesimal operators of the processes. Finally the author introduces the concept of hypermixing originating from large deviation theory. In this context also the notions of hyper- and ultracontractivity of Markov semigroups are discussed.\par The book represents an overview of the theory of mixing processes and fields which seems to be useful for researchers. In this respect it is a continuation and supplement of {\it E. Eberlein} and {\it M. S. Taqqu} (eds.) [Dependence in probability and statistics (Prog. Prob. Stat. 11, Birkhäuser, Boston, 1986)]. A lot of the results is proved or the proofs are sketched. Concerning linear fields, in Section 2.3 the proof of the Theorems 1 and 3 which generalize considerably a result of {\it V. V. Gorodetskij} [Theory Probab. Appl. 22(1977), 411-413 (1978); translation from Teor. Veroyatn. Primen. 22, 421-423 (1977; Zbl 0377.60046)] does not seem to be complete. Detailed references are given for all treated and even some omitted subjects.
[D.Tasche (Berlin)]
MSC 2000:
*60G05 Foundations of stochastic processes
60F05 Weak limit theorems
60-02 Research monographs (probability theory)
60F10 Large deviations

Keywords: weakly dependent; mixing coefficients; strong mixing; covariance inequalities; Rosenthal-type moment inequality; exponential inequalities; maximal inequalities; Gaussian fields; Gibbs fields; hypermixing; large deviation theory; hyper- and ultracontractivity of Markov semigroups

Citations: Zbl 0591.00012; Zbl 0377.60046

Cited in: Zbl 1067.60022 Zbl 0986.60042

45
Zbl 0790.60037
Doukhan, Paul; Massart, Pascal; Rio, Emmanuel
The functional central limit theorem for strongly mixing processes.
(English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 30, No.1, 63-82 (1994). ISSN 0246-0203

Summary: Let $(X\sb i)\sb{i\in\bbfZ}$ be a strictly stationary and strongly mixing sequence of $\bbfR\sp d$-valued zero-mean random variables. Let $(\alpha\sb n)\sb{n>0}$ be the sequence of mixing coefficients. We define the strong mixing function $\alpha$ by $\alpha(t)=\alpha\sb{[t]}$ and we denote by $Q$ the quantile function of $\vert X\sb 0\vert$, which is the inverse function of $t\to\bbfP(\vert X\sb 0\vert>t)$. The main result of this paper is that the functional central limit theorem holds whenever the following condition is fulfilled: $$\int\sp 1\sb 0\alpha\sp{- 1}(t)[Q(t)]\sp 2dt<\infty, \tag *$$ where $f\sp{-1}$ denotes the inverse of the monotonic function $f$. Note that this condition is equivalent to the usual condition $\bbfE(X\sp 2\sb 0)<\infty$ for $m$-dependent sequences. Moreover, for any $a>1$, we construct a sequence $(X\sb i)\sb{i\in\bbfZ}$ with strong mixing coefficients $\alpha\sb n$ of the order of $n\sp{-a}$ such that the CLT does not hold as soon as condition $(\sp*)$ is violated.
MSC 2000:
*60F17 Functional limit theorems
62G99 Nonparametric inference

Keywords: Donsker-Prokhorov invariance principle; strictly stationary; strongly mixing sequence; strong mixing function; functional central limit theorem

46
Zbl 0812.62043
Doukhan, Paul; León, José Rafael
Quadratic deviation of projection density estimates.
(English)
[J] REBRAPE 7, No.1, 37-63 (1993). ISSN 0103-0752

Authors' summary: We give here a technique of density estimation of projections generalizing that given by {\it N. N. Chentsov} [Sov. Math. Doklady 3, 1559-1562 (1963); translation from Doklady Akad. Nauk SSSR 147, 45-48 (1962; Zbl 0133.118)]. Simple assumptions imply an asymptotically Gaussian behaviour for the quadratic deviation of those estimates. Windows estimates given are optimal in some cases and a rate of convergence is given in the central limit result. These results are applied to compact Riemannian manifolds such as the multidimensional torus or sphere, to associated wavelet functions related to a multiscale analysis, to Hermite polynomials and to orthogonal polynomials such as Legendre polynomials.
[M.Denker (Göttingen)]
MSC 2000:
*62G07 Curve estimation
62G20 Nonparametric asymptotic efficiency
60F05 Weak limit theorems

Keywords: density estimation of projections; quadratic deviation; rate of convergence; central limit result; compact Riemannian manifolds; wavelet functions

Citations: Zbl 0133.118

47
Zbl 0804.62041
Doukhan, P.; Tsybakov, A.B.
Nonparametric recursive estimation in nonlinear ARX-models.
(English. Russian original)
[J] Probl. Inf. Transm. 29, No.4, 318-327 (1993); translation from Probl. Pereda. Inf. 29, No.4, 24-34 (1993). ISSN 0032-9460; ISSN 1608-3253

Summary: Consider the general $\text{ARX} (k,q)$ nonlinear process defined by the recurrence relation $$y\sb n= f(y\sb{n-1}, \dots, y\sb{n-k}, x\sb n,\dots, x\sb{n-q+1})+ \xi\sb n,$$ where $\{x\sb n\}$, $\{\xi\sb n\}$ are sequences of independent, identically distributed random variables. We propose a recursive nonparametric estimator of the function $f$ and we prove its strong consistency under general assumptions on the model. We study the model properties guaranteeing that these assumptions are satisfied.
MSC 2000:
*62G07 Curve estimation
62M10 Time series, etc. (statistics)

Keywords: general ARX(k,q) nonlinear process; recurrence relation; recursive nonparametric estimator; strong consistency

48
Zbl 0779.62072
Ango Nze, Patrick; Doukhan, Paul
(Nze, P.A.)
Functional estimation for mixing time series. (Estimation fonctionnelle de séries temporelles mélangeantes.)
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 317, No.4, 405-408 (1993). ISSN 0764-4442

Summary: Let $Z=(X\sb n,Y\sb n)\sb{n\in\bbfN\sp*}$ be a strongly mixing stationary stochastic process. We consider delta-estimates of the density of the marginal distribution of $X\sb 1$ and of the regression function $r(.)=\bbfE[Y\sb 1\vert X\sb 1=.]$ for kernel estimates. A finer evaluation of the variance of these estimates may be undertaken thanks to a new covariance inequality. The bounds reach an optimal order (that is the i.i.d.'s).\par Optimal bounds for MISE criterion are deduced from this basic result. We give uniform almost sure convergence results and uniform almost sure rates of convergence for such estimates. Uniform ${\cal L}\sp p$ bounds are also given. We give an outlook at both assumptions of strong dependence and absolute regularity. Minimax rates are attained.
MSC 2000:
*62M10 Time series, etc. (statistics)
62M09 Non-Markovian processes: estimation
62G07 Curve estimation
62G20 Nonparametric asymptotic efficiency
60G10 Stationary processes

Keywords: functional estimation; mixing time series; density estimation; uniform L(p) bounds; mean integrated square error; optimal bounds for MISE criterion; minimax rates; strongly mixing stationary stochastic process; delta-estimates; marginal distribution; regression function; kernel estimates; variance; new covariance inequality; uniform almost sure convergence results; uniform almost sure rates of convergence; strong dependence; absolute regularity

49
Zbl 0799.62041
Doukhan, Paul; Gassiat, Elisabeth
Quadratic deviation of penalized mean squares regression estimates.
(English)
[J] J. Multivariate Anal. 41, No.1, 89-101 (1992). ISSN 0047-259X

Consider the regression model $y\sb i = g(t\sb i) + \varepsilon\sb i$ and let $g\sb{n, \lambda}$ be a spline estimate of $g$ which minimizes the penalty functional $$n\sp{-1} \sum\sp n\sb 1 \bigl[ y\sb i - f(t\sb i) \bigr]\sp 2 + \lambda \int \bigl[ f\sp{(m)} (t) \bigr]\sp 2dt.$$ The paper presents a CLT for the quadratic functionals $$R\sb{n, \lambda} = \sum \bigl[ g\sb{n, \lambda} (t\sb i) - g(t\sb i) \bigr]\sp 2 \quad \text { and } \quad Z\sb{n, \lambda} = \int (g\sb{n, \lambda} - g)\sp 2dt$$ under various assumptions on the rate of $\lambda \to 0$ and behaviour of $g$. These statistics are suggested for goodness of fit testing purposes.\par The reviewer has two remarks: 1) The results on the asymptotic behaviour of $R\sb{n, \lambda}$ and $Z\sb{n, \lambda}$ under the hypothesis does not justify yet the use of these statistics for testing problems because under what kind of local alternatives it will be possible to reject remains unclear, 2) there are old papers by {\it V. D. Konakov} and {\it V. I. Piterbarg} [see Teor. Veroyatn. Primen. 27, No. 4, 707-724 (1982; Zbl 0503.60036), and ibid. 28, No. 1, 164-169 (1983; Zbl 0527.62047)] concerning $\sup \vert f\sb i - f \vert$ in density estimation problems which are closely related to what is studied in the present paper and are worthy of due reference.
MSC 2000:
*62G10 Nonparametric hypothesis testing
62G20 Nonparametric asymptotic efficiency
62G07 Curve estimation

Keywords: central limit theorem; spline estimate; penalty functional; quadratic functionals; goodness of fit testing

50
Zbl 0734.60054
Doukhan, Paul; Guyon, Xavier
Mixing of linear spatial random fields. (Mélange pour des processus linéaires spatiaux.)
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 313, No.7, 465-470 (1991). ISSN 0764-4442

Summary: We give sufficient strong mixing conditions for random fields $X=(X\sb t)\sb{t\in {\bbfZ}\sp d}$ defined linearly by $X\sb t=\sum\sb{s\in {\bbfZ}\sp d}g\sb{t,s}Z\sb s$, where Z is itself a mixing random field, extending a work by {\it V. V. Gorodetskij} [Theory Probab. Appl. 22(1977), 411-413 (1978); translation from Teor. Veroyatn. Primen. 22, 421-423 (1977; Zbl 0377.60046)] concerned with unilateral linear sequences. We give explicit bounds for those mixing coefficients, they usually depend on the cardinal of the subsets considered, at least for $d\ge 2$. We compare this case to the Gaussian one communicated by {\it I. A. Ibragimov} [Communication oracle, Orsay, 1991].
MSC 2000:
*60G60 Random fields

Keywords: strong mixing conditions; random fields; mixing random field

Citations: Zbl 0377.60046

51
Zbl 0734.60040
Doukhan, Paul; León, José R.
Estimation du spectre d'un processus gaussien fortement dépendant. (Spectral estimation for strongly dependent stationary Gaussian processes).
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 313, No.8, 523-526 (1991). ISSN 0764-4442

Summary: The empirical periodogram of a strongly dependent stationary Gaussian process with ${\bbfE} X\sb 0X\sb k\approx c\vert k\vert\sp{-\alpha}$ $(\alpha <1/2)$ satisfies for any $(1/2+\epsilon)$-Hölder continuous function g, if g(0)$\ne 0$, $n\sp{\alpha}(I\sb n(g)-E I\sb n(g))\to\sp{{\cal L}}g(0)Y$ for some non-Gaussian random variable Y, and, if fg is continuous, $\sqrt{n}(I\sb n(g)-{\bbfE} I\sb n(g))\to\sp{{\cal L}}N(0, 4\pi \int\sp{\pi}\sb{-\pi}f\sp 2(x)g\sp 2(x)dx).$ The spectral density f of the process is assumed to be continuous out of the origin. Considering a sequence $(g\sb{n,x})$ leads to estimates of f at a point x with analogous properties.
MSC 2000:
*60G10 Stationary processes
60B10 Convergence of probability measures
62M10 Time series, etc. (statistics)
60G15 Gaussian processes

Keywords: empirical periodogram; stationary Gaussian process; spectral density

52
Zbl 0719.60020
Bulinskii, Alexandre V.; Doukhan, Paul
(Bulinskij, A.V.)
Vitesse de convergence dans le théorème de limite centrale pour des champs mélangeants satisfaisant des hypothèses de moment faibles. (Rates in the central limit theorem for mixing random fields satisfying low moment assumptions).
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 311, No.12, 801-805 (1990). ISSN 0764-4442

Summary: We give an expression for the rate of convergence in the central limit theorem for mixing random fields assuming low moment assumptions, e.g. E $X\sp 2 \ln\sp{\delta}\sb+(X)<\infty$. We use, for it, a truncation in the first author's results [Sov. Math., Dokl. 34, No.3, 416-419 (1987); translation from Dokl. Akad. Nauk SSSR 291, 22-25 (1986; Zbl 0665.60028)] leading to results analogous to those well-known in the independent case [see {\it V. V. Petrov}, Limit theorems for the sums of independent random variables (1987; Zbl 0621.60022)].
MSC 2000:
*60F05 Weak limit theorems
60G60 Random fields

Keywords: central limit theorem; mixing random fields

Citations: Zbl 0665.60028; Zbl 0621.60022

53
Zbl 0702.60035
Doukhan, Paul; Léon, José Rafael
Déviation quadratique d'estimateurs de densité par projections orthogonales. (Quadratic deviation of projection density estimates).
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 310, No.6, 425-430 (1990). ISSN 0764-4442

Summary: We give here a technique of density estimation by projections generalizing that given by {\it N. N. Chentsov} [Sov. Math., Dokl. 3, 1559-1562 (1963; Zbl 0133.118); translation from Dokl. Akad. Nauk SSSR 147, 45-48 (1962)]. Simple assumptions imply an asymptotically Gaussian behaviour for quadratic deviation of those estimates. Window estimates are given optimal in some cases and a rate of convergence is given in the central limit result. Previous results are applied to compact Riemannian manifolds such as multidimensional torus or sphere, to associated wavelet functions associated to a multiscale analysis, to Hermite's polynomials and to orthogonal polynomials such as Legendre polynomials. An essential tool is a central limit theorem for increments of martingales adapted to our work.
MSC 2000:
*60F25 Lp-limit theorems (probability)
62G05 Nonparametric estimation
60G42 Martingales with discrete parameter

Keywords: technique of density estimation; compact Riemannian manifolds; Hermite's polynomials; increments of martingales

Citations: Zbl 0133.118

54
Zbl 0684.60027
Doukhan, P.; León, J.R.
Cumulants for stationary mixing random sequences and applications to empirical spectral density.
(English)
[J] Probab. Math. Stat. 10, No.1, 11-26 (1989). ISSN 0208-4147

A central limit theorem is derived for a strongly mixing stationary sequence under finiteness of cumulant sums and without any mixing rate assumption. Then a law of iterated logarithm is obtained. Further, the authors study the behaviour of empirical spectral density and present some applications of general results.
[J.Andel]
MSC 2000:
*60G10 Stationary processes
60F05 Weak limit theorems

Keywords: central limit theorem; strongly mixing stationary sequence; mixing rate; law of iterated logarithm; empirical spectral density

55
Zbl 0661.42019
Doukhan, Paul
Formes de Toeplitz associées à une analyse multi-échelle. (Toeplitz forms associated to a multiscale analysis).
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 306, No.15, 663-666 (1988). ISSN 0764-4442

Let but de ce travail est l'étude des formes de Toeplitz associées à une analyse multi-échelle de $L\sp 2({\bbfR}\sp d)$. Soit $\phi$ une fonction d'ondelette et soit $(V\sb j)\sb{j\in {\bbfN}}$ l'analyse graduée qui lui est associée. Une base inconditionnelle de $V\sb j$ est $(\Phi\sp j\sb h)\sb{h\in {\bbfZ}\sp d}$ où $\Phi\sp j\sb h(x)=2\sp{dj/2} \phi (2\sp jx\sb 1-h\sb 1)...\phi (2\sp jx\sb d-h\sb d).$ Soit $c\sp j\sb{h,\ell}=\int \Phi\sp j\sb h(x){\bar \Phi}\sp j\sb{\ell}(x)f(x)dx,$ la matrice de Toeplitz $T\sb j=(c\sp j\sb{h,\ell})\sb{\vert h\vert,\vert \ell \vert \le k(j)}$ est uniformément bornée lorsque $f\in L\sp{\infty}({\bbfR}\sp d)$, L'auteur s'intéresse à la distribution asymptotique des valeurs propres $\{\lambda\sp j\sb h$; $\vert h\vert \le k(j)\}$ de $T\sb j$ en considérant la mesure spectrale $\mu\sb j=2\sp{-dj}\sum\sb{\vert h\vert \le k}\delta\sb{\lambda\sp j\sb h}$. Il montre le comportement asymptotique des moments $M\sp s\sb{j,k}$ de $\mu\sb j$ quand j tend vers $+\infty$ et -$\infty$. En fait, supposant que $\vert \phi (x)\vert \le C(1+\vert x\vert)\sp{-q-1}$ avec $q>d$ si $s\ne 1$, $q>0$ sinon, et f, $Df\in L\sp 1\cap L\sp{\infty}$, alors si $k=k(j)$ est tel que $\lim\sb{j\to \infty}k2\sp{-j}=+\infty$, $\lim\sb{j\to \infty}k\sp d 2\sp{-(1+d)j}=0,$ on a $\lim\sb{j\to \infty}M\sp s\sb{j,k}=b\sp{*s}(0)\int f\sp s(x)dx,$ où $b(h)=\int \Phi (x){\bar \Phi}(x+h)dx.$ Alors si $\phi$ est une ondelette associée orthogonale, $\lim\sb{j\to \infty}\mu\sb j(g)=\int g(f(x))dx$ pour $g\in \Gamma =\{g\in C({\bbfR})$; h est à croissance polynomiale et $h(x)=x\sp{- 1}g(x)\}$. Si $\vert \phi (x)\vert +\vert \phi '(x)\vert \le C(1+\vert x\vert)\sp{-q-1}$ pour un $q>d-1$ et si $\int (\vert x\vert +1)\vert f(x)\vert dx<\infty,$ alors $\lim\sb{j\to \infty}\nu\sb j(g)=g(A\int f(x)dx),$ où $g\in \Gamma$, $A=\sum\sb{h\in {\bbfZ}\sp d}\vert \Phi (h)\vert\sp 2$ et où $\nu\sb j=2\sp{dj}\sum\sb{\vert h\vert \le k}\delta\sb{\lambda\sb h\sp{-j}}.$
[J.Ludwig]
MSC 2000:
*42C99 Non-trigometric Fourier analysis

Keywords: Toeplitz forms; multiscale analysis; Toeplitz matrix; wavelet function

56
Zbl 0964.62503
Doukhan, Paul
Nonparametric estimation for mixing random sequences.
(English)
[J] Acta Cient. Venez. 38, No.5-6, 585-590 (1987).

The author gives a review of various nonparametric estimates for mixing sequences of random variables. He discusses mainly different forms of kernel and projection estimates. For such estimates of densities and regression functions the integrated risk is studied in more detail.
[Friedrich Liese (Rostock)]
MSC 2000:
*62G07 Curve estimation
62G08 Nonparametric regression
57
Zbl 0659.60009
Bulinskii, Alexandre; Doukhan, Paul
Inégalités de mélange fort utilisant des normes d'Orlicz. (Strong mixing moment inequalities using Orlicz norms).
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 305, 827-830 (1987). ISSN 0764-4442

We first extend the Orlicz inequality for covariances given by the first author [On strong mixing conditions for random fields and central limit theorem, Sov. Math., Dokl. (to appear)] to the Hilbert valued case. After this we show generalizations of the Marcinkiewicz-Zygmund inequality of higher order moments for sums of strongly mixing random variables extending to Orlicz case results by the second author and {\it F. Portal} [ibid. 297, Sér. I, 129-132 (1983; Zbl 0544.62022)] and the second author, {\it J. Léon} and {\it F. Portal} [ibid. 298, Sér. I, 305-308 (1984; Zbl 0557.60006)]. Orlicz norms allow weakening of moment assumptions. Interest of the results is explicited by the example of kernel density estimates.
MSC 2000:
*60B05 Probability measures on topological spaces
60E15 Inequalities in probability theory

Keywords: Orlicz inequality; Marcinkiewicz-Zygmund inequality; Orlicz norms

Citations: Zbl 0544.62022; Zbl 0557.60006

58
Zbl 0651.60042
Doukhan, Paul; Portal, Frederic
Principe d'invariance faible pour la fonction de repartition empirique dans un cadre multidimensionnel et melangeant. (Weak invariance principle for multivariate and mixing empirical distributions).
(French)
[J] Probab. Math. Stat. 8, 117-132 (1987). ISSN 0208-4147

Let $\{\xi\sb n,\ n\ge0\}$ be a strictly stationary $R\sp d$-valued process. Suppose $d=1$. The authors obtain inequalities of Marcinkiewicz- Zygmund type for even order moments of partial sums of the process $\{\xi\sb n\}$ assuming that it is either a strong-mixing or a uniformly mixing ($\varphi$-mixing) process. They also derive an exponential inequality of Bernstein-type when the process is geometrically $\varphi$- mixing, that is, the mixing coefficient $\phi\sb n\le a\theta\sp n$, $a\ge0$, $0\le\theta<1$.\par Assume that $d\ge1$. Suppose $F\sb n$ denotes the empirical distribution function corresponding to $\{\xi\sb1,\ldots,\xi\sb n\}$ under the usual partial ordering on $R\sp d$ and $\{\xi\sb n\}$ is geometrically $\varphi$-mixing. Let $F$ be the distribution function of $\xi\sb1.$\par The authors prove that there exists a sequence of zero-mean stationary Gaussian processes $Y\sb n$ with appropriate covariance function such that $$P\left\{ \sup\sb{R\sp d}\left\vert \sqrt n (F\sb n - F) - Y\sb n \right\vert \ge cn\sp{-a}\log n \right\} \le cn\sp{-a}\log n$$ where $a=1/(3(5d+4))$ and $c$ is a constant depending on $d$ and $\theta$. Similar results were obtained for strong-mixing and $\varphi$-mixing processes.
[B.L.S.Prakasa Rao]
MSC 2000:
*60F17 Functional limit theorems
60G15 Gaussian processes

Keywords: inequalities of Marcinkiewicz-Zygmund type; strong-mixing; uniformly mixing; exponential inequality; phi-mixing processes

59
Zbl 0596.60037
Doukhan, P.; Leon, J.R.; Portal, F.
Principes d'invariance faible pour la mesure empirique d'une suite de variables aléatoires mélangeante. (Weak invariance principles for the empirical measure of a mixing sequence of random variables).
(French)
[J] Probab. Theory Relat. Fields 76, 51-70 (1987). ISSN 0178-8051; ISSN 1432-2064

We give weak invariance principles for the empirical measure of a stationary strongly mixing sequence $(\xi\sb k)\sb{k\ge 0}$, $$X\sb n(f)=(1/\sqrt{n})\sum\sp{n}\sb{k=1}(f(\xi\sb k)-Ef(\xi\sb k)).$$ For the case where $f\in B\sb s$, the unit ball of the Sobolev space $H\sb s(X)$ of a Riemannian compact manifold and f is a $Lip\sb{\alpha}$ function $(<\alpha \le 1)$, we obtain logarithmic rates of convergence $\epsilon\sb n$ such that, for a stationary sequence of Gaussian processes $Y\sb n$, $${\bbfP}(\sup \vert X\sb n(f)-Y\sb n(f)\vert >\epsilon\sb n)\le \epsilon\sb n.$$ We also prove, for the case of kernel estimates $\hat g\sb n$, the existence of a Gaussian nonstationary sequence of random processes $(Y\sb n(x))\sb{x\in K}$ indexed be a compact subset K of ${\bbfR}\sp d$ and constants $a,b,c>0$ such that $${\bbfP}(\sup\sb{x\in K}\vert n h\sp d\sb n)\sp{1/2} (\hat g\sb n(x)- g(x))-Y\sb n(x)\vert \ge c\quad n\sp{-a})\le cn\sp{-a}\quad if\quad h\sb n=n\sp{-b};$$ finally we give estimates of the kind: $E\sup\sb{x\in K}(\hat g\sb n(x)-g(x))\sp 2\le cn\sp{-a}$ if $h\sb n=n\sp{-b}$. Here $h\sb n$ is the window of the kernel estimate $\hat g\sb n$.
MSC 2000:
*60F17 Functional limit theorems
60G15 Gaussian processes
62G05 Nonparametric estimation

Keywords: weak invariance principles; empirical measure; logarithmic rates of convergence; kernel estimate

60
Zbl 0613.62050
Doukhan, Paul
Fonctions d'Hermite et statistiques des processus melangeants. (Hermite functions and statistics of mixing processes).
(French)
[J] Cah. Cent. Étud. Rech. Opér. 28, 99-115 (1986). ISSN 0774-3068

In this paper orthogonal series estimates based on Hermite functions for probability densities and regression functions and their derivatives are investigated. Several measures of deviation, e.g. the MISE on ${\bbfR}$ or the expected uniform deviation on a compact interval are explored. \par In case of independent samples e.g. {\it S. C. Schwartz} [Ann. Math. Stat. 38, 1261-1265 (1967; Zbl 0157.479)], {\it G. G. Walter} [Ann. Stat. 5, 1258-1264 (1977; Zbl 0375.62041)] or {\it W. Greblicki} and {\it M. Pawlak} [J. Multivariate Anal. 15, 174-182 (1984; Zbl 0544.62043)] have evaluated the MISE, whereas in this paper stationary $\phi$- and strongly mixing samples are admitted too. Furthermore e.g. a central limit theorem for the $L\sb 2$-deviation of the estimates, the behavior of a kernel estimate related to the Hermite-system (Mehler kernel) and a weak invariance principle for empirical distributions with an index space containing functions with small Fourier coefficients are presented as well.
MSC 2000:
*62G05 Nonparametric estimation
33C45 Orthogonal polynomials and functions of hypergeometric type
60F05 Weak limit theorems

Keywords: quadratic deviation; quadratic uniform risk; density estimation; asymptotic normality; phi-mixing; orthogonal series estimates; Hermite functions; measures of deviation; MISE; expected uniform deviation; strongly mixing; central limit theorem; Mehler kernel; weak invariance principle; empirical distributions

Citations: Zbl 0157.479; Zbl 0375.62041; Zbl 0544.62043

61
Zbl 0599.62043
Doukhan, P.; Leon, J.; Portal, F.
Une mesure de la déviation quadratique d'estimateurs non paramétriques. (A measure of quadratic deviation of nonparametric estimators).
(French)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 22, 37-66 (1986). ISSN 0246-0203

Let $\theta$ be either the density f of $U\sb 1$ or the regression function $r=E(V\sb 1\vert U\sb 1)$ or the product rf and let ${\hat \theta}{}\sb n$ be a kernel-type or an orthogonal series estimator of $\theta$ based on n identically distributed ${\bbfR}\sp d$-valued random variables $(U\sb i,V\sb i)\sb{1\le i\le n}$. Nonrandom sequences $(a\sb n)\sb{n\in {\bbfN}\sp*}$ and $(b\sb n)\sb{n\in {\bbfN}\sp*}$ are determined such that the statistics $$(a\sb n\int \vert {\hat \theta}\sb n-\theta \vert d\mu -b\sb n)\sb{n\in {\bbfN}\sp*},$$ converge to a Gaussian distribution $N(0,\sigma\sp 2)$ ($\mu$ is positive, $\sigma$- finite and absolutely continuous w.r.t. the Lebesgue measure). Both independent and mixing case are dealt with. It must be noticed that $a\sb n$, $b\sb n$ and $\sigma$ do not depend on the mixing function. The basic tools of the paper are Gaussian approximations in Hilbert spaces and the Karhunen-Loeve expansion.
[A.Berlinet]
MSC 2000:
*62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics

Keywords: kernel-type estimator; density estimation; nonparametric regression; mean square error; asymptotic normality; orthogonal series estimator; Gaussian distribution; mixing; Gaussian approximations; Hilbert spaces; Karhunen- Loeve expansion

62
Zbl 0594.60040
Doukhan, P.; Leon, J.R.
Invariance principles for the empirical measure of a mixing sequence and for the local time of Markov processes.
(English)
[A] Geometrical and statistical aspects of probability in Banach spaces, Proc. Conf., Strasbourg/France 1985, Lect. Notes Math. 1193, 4-21 (1986).

[For the entire collection see Zbl 0581.00015.] \par The paper consists of two parts. The first one is devoted to an investigation of the rate of convergence in the weak invariance principle for the empirical process $$X\sb n(f)=n\sp{- 1/2}\sum\sp{n}\sb{k=1}(f(\xi\sb k)-Ef(\xi\sb k)),\quad f\in B\sb s.$$ Here $\{\xi\sb k\}$ is a strictly stationary strongly mixing sequence with values in a d-dimensional Riemannian compact manifold E and $B\sb s$ is the unit ball of the Sobolev space $H\sb s$ for $s>d/2.$ \par The second part studies the asymptotic behaviour of $$Z\sb n(f)=n\sp{- 1/2}\int\sp{n}\sb{0}f(X\sb u)du,\quad f\in L\sp 2(\mu),$$ where $\{X\sb t$, $t>0\}$ is a recurrent ergodic stationary Markov process with values in a compact Riemannian manifold E and $\mu$ is the invariant measure of the process. An invariance principle in a general framework is proved. For $X\sb t$ being the Brownian motion a law of iterated logarithm uniform on the unit ball $B\sb s$ of $H\sb s$ for $s>d/2-1$ is obtained. The case of diffusions as well as a discretization of $Z\sb n$ are also considered. \par The paper is related to some earlier results of {\it J. R. Baxter} and {\it G. A. Brosamler} [Math. Scand. 38, 115-136 (1976; Zbl 0346.60020)], {\it E. Bolthausen} [Stochastic Processes Appl. 16, 199-204 (1984; Zbl 0524.60024)], {\it D. Dacunha-Castelle} and {\it D. Florens-Zmirou} [C. R. Acad. Sci., Paris, Sér. I 299, 65-68 (1984; Zbl 0579.60044)] and others.
[T.Inglot]
MSC 2000:
*60F17 Functional limit theorems
60F15 Strong limit theorems
60J25 Markov processes with continuous parameter
60J60 Diffusion processes

Keywords: rate of convergence in the weak invariance principle; empirical process; strongly mixing sequence; Riemannian manifold

63
Zbl 0607.60019
Doukhan, P.; Leon, J.; Portal, F.
Calcul de la vitesse de convergence dans le théorème central limite vis a vis des distances de Prohorov, Dudley et Levy dans le cas de variables aléatoires depéndantes. (Speed of convergence in the central limit theorem with respect to Prokhorov, Dudley and Levy distances in the case of dependent random variables).
(French)
[J] Probab. Math. Stat. 6, 19-27 (1985). ISSN 0208-4147

The paper gives a general framework to estimate Dudley and Lévy's metrics for Hilbert space valued random variables and Prohorov's one for the k-dimensional distributions of an ${\bbfR}\sp d$-valued process, in the case of central limit theorem for stationary and mixing random variables. The speeds of convergence obtained here are approximately $n\sp{-1/4}$, $n\sp{-1/12}$ and $k\sp{5/8}n\sp{-1/12}$, where n is the length of the observed sample and with quite strong mixing hypotheses.
MSC 2000:
*60F05 Weak limit theorems
60G10 Stationary processes

Keywords: speed of convergence; central limit theorem; Dudley and Lévy's metrics; stationary and mixing random variables; strong mixing hypotheses

64
Zbl 0557.60006
Doukhan, Paul; Leon, José; Portal, Frédéric
Vitesse de convergence dans le théorème central limite pour des variables aléatoires mélangeantes à valeurs dans un espace de Hilbert. (Speed of convergence in the central limit theorem for mixing Hilbert space valued random variables.).
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 298, 305-308 (1984). ISSN 0764-4442

Summary: We consider random variables with values in a separable Hilbert space which constitute a $\phi$-mixing sequence or a strongly mixing field indexed by ${\bbfZ}\sp d$. We show Marcinkiewicz-Zygmund inequalities and exponential inequalities where the random variables are bounded and mixing is geometric. We also show Berry-Essen estimates for the central limit theorem in the stationary case; our estimates are of order $n\sp{\eta -\gamma}$ with $\gamma =1/4(2+d)$ for $d\le 2$, and $\gamma =1/7d$ else, $\eta$ is arbitrary little and n is the number of random variables.
MSC 2000:
*60B12 Limit theorems for vector-valued random variables (inf.-dim.case)
60F05 Weak limit theorems

Keywords: strongly mixing field; Marcinkiewicz-Zygmund inequalities; exponential inequalities; Berry-Essen estimates; central limit theorem

Cited in: Zbl 0659.60009

65
Zbl 0544.62022
Doukhan, Paul; Portal, Frédéric
Moments de variables aléatoires mélangeantes.
(English)
[J] C. R. Acad. Sci., Paris, Sér. I 297, 129-132 (1983). ISSN 0764-4442

This note proves several majorization inequalities for moments of sums of $\phi$-mixing or strongly mixing (but not necessarily stationary) random variables. The authors also briefly discuss applications to empirical distribution functions for mixing sequences and to kernel estimators.
[E.Slud]
MSC 2000:
*62E99 Statistical distribution theory
60E15 Inequalities in probability theory
62G30 Order statistics, etc.
62G05 Nonparametric estimation
60G12 General second order processes

Keywords: Khinchin-Kahane type inequalities; phi mixing sequences; majorization inequalities; moments of sums; strongly mixing; empirical distribution functions; kernel estimators

Cited in: Zbl 0659.60009

66
Zbl 0529.62037
Collomb, Gerard; Doukhan, Paul
Estimation non paramètrique de la fonction d'autoregression d'un processus stationnaire et phi melangeant: risques quadratiques pour la méthode du noyau.
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 296, 859-862 (1983). ISSN 0764-4442
MSC 2000:
*62G05 Nonparametric estimation
62M10 Time series, etc. (statistics)

Keywords: stationary phi-mixing process; kernel regression estimates; majorization; quadratic risks; rate of convergence of estimates; autoregressive processes

67
Zbl 0529.60029
Doukhan, Paul; Portal, Frederic
Principe d'invariance faible avec vitesse pour un processus empirique dans un cadre multidimensionnel et fortement melangeant.
(French)
[J] C. R. Acad. Sci., Paris, Sér. I 297, 505-508 (1983). ISSN 0764-4442
MSC 2000:
*60F17 Functional limit theorems

Keywords: weak invariance principle; empirical process; Prohorov distance; rate of convergence; mixing rate

Cited in: Zbl 0737.62077

68
Zbl 0515.62037
Doukhan, Paul; Ghindes, Marcel
Estimation de la transition de probabilité d'une chaîne de Markov Doeblin-recurrente. Étude du cas du processus autoregressif général d'ordre 1.
(French)
[J] Stochastic Processes Appl. 15, 271-293 (1983). ISSN 0304-4149

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MSC 2000:
*62G05 Nonparametric estimation
62M05 Markov processes: estimation

Keywords: homogeneous Markov chain; density estimations; kernel estimators; regression function; autoregressive model; Doeblin recurrent; transition probability estimation; integrated mean square error; uniform convergence

69
Zbl 0503.62085
Doukhan, Paul
Simulation in the general first order autoregressive process (unidimensional normal case).
(English)
[A] Specifying statistical models, Proc. 2nd Franco-Belgian Meet. Stat., Louvain-la-Neuve/Belg. 1981, Lect. Notes Stat. 16, 50-68 (1983).

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MSC 2000:
*62M10 Time series, etc. (statistics)
62G05 Nonparametric estimation
65C99 Numerical simulation
65C05 Monte Carlo methods

Keywords: first order autoregressive process; unidimensional normal case; kernel estimators; invariant measures

Citations: Zbl 0497.00014

70
Zbl 0461.62070
Doukhan, Paul; Ghindes, Marcel
Estimations dans le processus "X//(n+1)=f($X\sb n)$+epsilon//n".
(French)
[J] C. R. Acad. Sci., Paris, Sér. A 291, 61-64 (1980).

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MSC 2000:
*62M05 Markov processes: estimation

Keywords: densities of transition probability; density estimations; density of invariant measure

71
Zbl 0433.60069
Doukhan, Paul; Ghindes, Marcel
Étude du processus: $X_{n+1}=f(X_n)\cdot\varepsilon_n$".
(French)
[J] C. R. Acad. Sci., Paris, Sér. A 290, 921-923 (1980).

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MSC 2000:
*60J05 Markov processes with discrete parameter

Keywords: Markov chain; geometric ergodicity

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