Empirical Process Techniques for Dependent Data

Abstract

Let (X _k) _k≥1 be a sequence of random variables with common distribution function F(x) = P(X ₁ ≤ x). Define the empirical distribution function

$$ {{F}_{n}}(x) = \frac{1}{n}\# \{ 1 \leqslant i \leqslant n:{{X}_{1}} \leqslant x\} , $$

and the empirical process by $\sqrt {n} ({{F}_{n}}(x) - F(x))$ In this chapter we provide a survey of classical as well as modern techniques in the study of empirical processes of dependent data. We begin with a sketch of the early roots of the field in the theory of uniform distribution mod 1, of sequences defined by X _k = {n _k ω}, ω ∈ [0, 1], dating back to Weyl’s celebrated 1916 paper. In the second section we provide the essential tools of empirical process theory, and we prove Donsker’s classical empirical process invariance principle for i.i.d. processes. The third section provides an introduction to the subject of weakly dependent random variables. We introduce a variety of mixing concepts, provide necessary technical tools like correlation and moment inequalities, and prove central limit theorems for partial sums. The empirical process of weakly dependent data is investigated in the fourth section, where we put special emphasis on almost sure approximation techniques. The fifth section is devoted to the empirical distribution of U-statistics, defined as

$$ Un(x) = {{\left( {\begin{array}{*{20}{c}} n \\ 2 \\ \end{array} } \right)}^{{ - 1}}}\# \{ 1 \leqslant i < j \leqslant n:h({{X}_{i}},{{X}_{j}}) \leqslant x\} $$

for some symmetric kernel h. We give some applications, e.g., to dimension estimation in the analysis of time series, and prove weak convergence of the corresponding empirical process. Empirical processes of long-range dependent data are the topic of the sixth section. We give an introduction to the area of long-range dependent processes, provide important technical tools for the study of their partial sums and investigate the limit behavior of the empirical process. It turns out that the limit process is of a completely different type as in the case of independent or weakly dependent data, and that this has important consequences for various functionals of the empirical process. The final section is devoted to pair correlations, i.e., U-statistics empirical processes over short intervals associated with the kernel h(x, y) = |x − y|