On the functional central limit theorem via martingale approximation

Details On the functional central limit theorem via martingale approximation On the functional central limit theorem via martingale approximation

2009-10-19

arxiv.org/abs/0910.3448v3

Bernoulli 2011, Vol. 17, No. 1, 424-440

Mathematics

Probability

Published in at http://dx.doi.org/10.3150/10-BEJ276 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/10-BEJ276

In this paper, we develop necessary and sufficient conditions for the validity of a martingale approximation for the partial sums of a stationary process in terms of the maximum of consecutive errors. Such an approximation is useful for transferring the conditional functional central limit theorem from the martingale to the original process. The condition found is simple and well adapted to a variety of examples, leading to a better understanding of the structure of several stochastic processes and their asymptotic behaviors. The approximation brings together many disparate examples in probability theory. It is valid for classes of variables defined by familiar projection conditions such as the Maxwell--Woodroofe condition, various classes of mixing processes, including the large class of strongly mixing processes, and for additive functionals of Markov chains with normal or symmetric Markov operators.

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