Functional Limit Theorem for the Empirical Process of a Class of Bernoulli Shifts with Long Memory

Paul Doukhan, Gabriel Lang, Donatas Surgailis and Marie-Claude Viano
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Paul Doukhan: LS-CREST and University Cergy Pontoise
Gabriel Lang: Laboratoire GRESE
Donatas Surgailis: Vilnius Institute of Mathematics and Informatics
Marie-Claude Viano: Laboratoire de Mathématiques Appliquées

Journal of Theoretical Probability, 2005, vol. 18, issue 1, 161-186

Abstract: We prove a functional central limit theorem for the empirical process of a stationary process X t =Y t +V t , where Y t is a long memory moving average in i.i.d. r.v.’s ζ s , s ≤ t, and V t =V (ζ t , ζt-1,...) is a weakly dependent nonlinear Bernoulli shift. Conditions of weak dependence of V t are written in terms of L2-norms of shift-cut differences V (ζ t , ζt-n, 0,...,) − V(ζ t ,...,ζt-n+1, 0,...). Examples of Bernoulli shifts are discussed. The limit empirical process is a degenerated process of the form f(x)Z, where f is the marginal p.d.f. of X0 and Z is a standard normal r.v. The proof is based on a uniform reduction principle for the empirical process.

Keywords: Functional limit theorems; self-similar process; times series; 60F17; 60G18; 62M10 (search for similar items in EconPapers)
Date: 2005
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DOI: 10.1007/s10959-004-2593-3

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