Central Limit Theorems for Randomly Modulated Sequences of Random Vectors with Resampling and Applications to Statistics
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Limit Theorems for Random Fields
Limit Theorems for Random Fields A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematical Sciences College of Arts and Sciences University of Cincinnati, July 2019 Author: Na Zhang Chair: Magda Peligrad, Ph.D. Degrees: B.S. Mathematics, 2011, Committee: Wlodzimierz Bryc, Ph.D. Huaibei Normal University Yizao Wang, Ph.D. M.S. Financial Mathematics, 2014, Soochow University Abstract The focus of this dissertation is on the dependent structure and limit theorems of high dimensional probability theory. In this dissertation, we investigate two related topics: the Central Limit Theorem (CLT) for stationary random fields (multi-indexed random variables) and for Fourier transform of stationary random fields. We show that the CLT for stationary random processes under a sharp projective condition introduced by Maxwell and Woodroofe in 2000 [MW00] can be extended to random fields. To prove this result, new theorems are established herein for triangular arrays of martingale differences which have interest in themselves. Later on, to exploit the richness of martingale techniques, we establish the necessary and sufficient conditions for martingale approximation of random fields, which extend to random fields many corresponding results for random sequences (e.g. [DMV07]). Besides, a stronger form of convergence, the quenched convergence, is investigated and a quenched CLT is obtained under some projective criteria. The discrete Fourier transform of random fields, (Xk)k2Zd (d ≥ 2), where Z is the set of integers, is defined as the rotated sum of the random fields. Being one of the important tools to prove the CLT for Fourier transform of random fields, the law of large numbers (LLN) is obtained for discrete Fourier transform of random sequences under a very mild regularity condition. -
Chapter 6 Stationary Stochastic Processes
Chapter 6 Stationary Stochastic Processes. 6.1 Ergodic Theorems. A stationary stochastic process is a collection {ξn : n ∈ Z} of random vari- ables with values in some space (X, B) such that the joint distribution of ··· ··· (ξn1 , ,ξnk ) is the same as that of (ξn1+n, ,ξnk+n) for every choice of k ≥ 1, and n, n1, ··· ,nk ∈ Z. Assuming that the space (X, B) is reasonable and Kolmogorov’s consistency theorem applies, we can build a measure P on the countable product space Ω of sequences {xn : n ∈ Z} with values in X, defined for sets in the product σ-field F. On the space Ω there is the natural shift defined by (Tω)(n)=xn+1 for ω with ω(n)=xn. The random variables xn(ω)=ω(n) are essentially equivalent to {ξn}. The stationarity of the pro- cess is reflected in the invariance of P with respect to T i.e. PT−1 = P .We can without being specific consider a space Ω a σ-field F, a one to one invert- ible measurable map from Ω → Ω with a measurable inverse T −1 and finally a probability measure P on (Ω, F)thatisT -invariant i.e P (T −1A)=P (A) for every A ∈F. One says that P is an invariant measure for T or T is a measure preserving transformation for P . If we have a measurable map from n ξ :(Ω, F) → (X, B), then it is easily seen that ξn(ω)=ξ(T ω) defines a sta- tionary stochastic process. The study of stationary stochastic process is then more or less the same as the study of measure preserving (i.e. -
Recent Advances in Invariance Principles for Stationary Sequences 2
Probability Surveys Vol. 3 (2006) 1–36 ISSN: 1549-5787 DOI: 10.1214/154957806100000202 Recent advances in invariance principles for stationary sequences∗ Florence Merlev`ede Universit´eParis 6, LPMA and C.N.R.S UMR 7599, 175 rue du Chevaleret, 75013 Paris, FRANCE e-mail: [email protected] Magda Peligrad† Department of Mathematical Sciences, University of Cincinnati, PO Box 210025, Cincinnati, Oh 45221-0025, USA e-mail: [email protected] Sergey Utev School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, ENGLAND e-mail: [email protected] Abstract: In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main tool for obtaining the invariance principles, and also they have interest in themselves. The classes of dependent random variables considered will be martingale-like sequences, mixing sequences, linear processes, additive functionals of er- godic Markov chains. AMS 2000 subject classifications: 60G51, 60F05. Keywords and phrases: Brownian motion, weakly dependent sequences, martingale differences, linear process, central limit theorem, invariance prin- ciple. Received November 2004. Contents 1 Introduction 2 arXiv:math/0601315v2 [math.PR] 24 Feb 2006 2 Definitions and notations 3 3 Maximal inequalities 5 ∗This is an original survey paper †Supported in part by a Charles Phelps Taft Memorial Fund grant and NSA grant H98230- 05-1-0066. 1 F. Merlev`ede et al./Recent advances in invariance principles for stationary sequences 2 4 Invariance principles 8 4.1 The normalization √n ........................ 8 4.2 Examples.Part1.......................... -
Central Limit Theorem and Almost Sure Results for the Empirical Estimator of Superquantiles/Cvar in the Stationary Case Jérôme Dedecker, Florence Merlevède
Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case Jérôme Dedecker, Florence Merlevède To cite this version: Jérôme Dedecker, Florence Merlevède. Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case. 2021. hal-03214790 HAL Id: hal-03214790 https://hal.archives-ouvertes.fr/hal-03214790 Preprint submitted on 2 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Central limit theorem and almost sure results for the empirical estimator of superquantiles/CVaR in the stationary case J´er^omeDedecker,∗ Florence Merlev`ede y May 2, 2021 Abstract In this paper, we show that the difference between the empirical estimator and the Conditional value-at-risk can be written as a simple partial sum + a residual term. Starting from this decomposition, we prove a central limit theorem and some almost sure results for the empirical estimator, for a large class of stationary se- quences. We also construct a confidence interval with asymptotic level 1 − α, and we study its coverage level through two different sets of simulation. Mathematics subject classifications. -
Split Invariance Principles for Stationary Processes
Split invariance principles for stationary processes István Berkes1, Siegfried Hörmann and Johannes Schauer2 Graz University of Technology, Université Libre de Bruxelles and Graz University of Technology Abstract The results of Komlós, Major and Tusnády give optimal Wiener approx- imation of partial sums of i.i.d. random variables and provide an extremely powerful tool in probability and statistical inference. Recently Wu [52] ob- tained Wiener approximation of a class of dependent stationary processes with finite p-th moments, 2 < p · 4, with error term o(n1=p(log n)γ), γ > 0, and Liu and Lin [27] removed the logarithmic factor, reaching the Komlós-Major- Tusnády bound o(n1=p). No similar results exist for p > 4 and in fact, no existing method for dependent approximation yields an a.s. rate better than o(n1=4). In this paper we show that allowing a second Wiener component in the approximation, we can get rates near to o(n1=p) for arbitrary p > 2. This extends the scope of applications of the results essentially, as we illustrate it by proving new limit theorems for increments of stochastic processes and sta- tistical tests for short term (epidemic) changes in stationary processes. Our method works under a general weak dependence condition similar to those in [27] and [52] covering, among others, wide classes of linear and nonlinear time series models and classical dynamical systems. Keywords: Stationary processes, strong invariance principle, KMT approximation, dependence, increments of partial sums MSC 2000: 60F17, 60G10, 60G17 1)Research supported by FWF grant S9603-N23 and OTKA grants K 67961 and K 81928 2)Research supported by FWF grant S9603-N23 1 1 Introduction Let X; X1;X2;::: be i.i.d. -
Strong Approximation of the Empirical Distribution Function for Absolutely Regular Sequences in Rd
Strong approximation of the empirical distribution function for absolutely regular sequences in Rd. 08/03/2013 J´er^omeDedecker a, Florence Merlev`ede b and Emmanuel Rio c. a Universit´eParis Descartes, Laboratoire MAP5, UMR 8145 CNRS, 45 rue des Saints-P`eres,F-75270 Paris cedex 06, France. E-mail: [email protected] b Universit´eParis-Est, LAMA (UMR 8050), UPEMLV, CNRS, UPEC, F-77454 Marne-La-Vall´ee,France. E-mail: fl[email protected] b Universit´ede Versailles, Laboratoire de math´ematiques,UMR 8100 CNRS, B^atiment Fermat, 45 Avenue des Etats-Unis, 78035 Versailles, FRANCE. E-mail: [email protected] Running head: Strong approximation for the empirical process. Key words: Strong approximation, Kiefer process, empirical process, stationary sequences, absolutely regular sequences. Mathematical Subject Classification (2010): 60F17, 60G10. Abstract We prove a strong approximation result with rates for the empirical process associated to an ab- solutely regular stationary sequence of random variables with values in Rd. As soon as the absolute regular coefficients of the sequence decrease more rapidly than n1−p for some p 2]2; 3], we show that the error of approximation between the empirical process and a two-parameter Gaussian process is of order n1=p(log n)λ(d) for some positive λ(d) depending on d, both in L1 and almost surely. The power of n being independent of the dimension, our results are even new in the independent setting, and improve earlier results. In addition, for absolutely regular sequences, we show that the rate of approximation is optimal up to the logarithmic term.