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An introduction to functional central limit theorems for dependent stochastic processes. (English) Zbl 0834.60033
The paper has the form of a survey on the functional central limit theorems for dependent stochastic processes and in particular for empirical processes \[ \nu_n f = n^{-1/2} \sum^n_{i = 1} \bigl( f(\xi_{ni}) - Ef (\xi_{ni}) \bigr), \] where \(f\) is a function from a class of functions and \(\{\xi_{ni} : i \leq n, n = 1,2, \ldots\}\) is a strong mixing triangular array. The proofs are based on the moment inequality and the notion of stochastic equicontinuity which is more useful, as the authors remark, than the notion of the uniform tightness for a set of probability measures. The authors write in their summary that “this paper shows how the modern machinery for generating abstract empirical central limit theorems can be applied to arrays of dependent variables. It develops a bracketing approximation (closely related to results of Philipp and Massart) based on a moment inequality for sums of strong mixing arrays, in an affort to illustrate the sort of difficulty that need to be overcome when adapting the empirical process theory for independent variables”.

60F17 Functional limit theorems; invariance principles
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