In this paper we present a tail inequality for the maximum of partial sums of a weakly dependent sequence of random variables that is not necessarily bounded. The class considered includes geometrically and subgeometrically strongly mixing sequences. The result is then used to derive asymptotic moderate deviation results. Applications include classes of Markov chains, functions of linear processes with absolutely regular innovations and ARCH models.
Following Lindeberg's approach, we obtain a new condition for a stationary sequence of square-integrable and real-valued random variables to satisfy the central limit theorem. In the adapted case, this condition is weaker than any projective criterion derived from Gordin's theorem [Dokl. Akad. Nauk SSSR 188 (1969) 739-741] about approximating martingales. Moreover, our criterion is equivalent to the conditional central limit theorem, which implies stable convergence (in the sense of Rényi) to a mixture of normal distributions. We also establish functional and triangular versions of this theorem. From these general results, we derive sufficient conditions which are easier to verify and may be compared to other results in the literature. To be complete, we present an application to kernel density estimators for some classes of discrete time processes.
In this paper we obtain a Bernstein type inequality for a class of weakly dependent and bounded random variables. The proofs lead to a moderate deviations principle for sums of bounded random variables with exponential decay of the strong mixing coefficients that complements the large deviation result obtained by Bryc and Dembo (1998) under superexponential mixing rates.
In this paper, we obtain almost sure invariance principles with rate of order n 1/p log β n, 2 < p ≤ 4, for sums associated to a sequence of reverse martingale differences. Then, we apply those results to obtain similar conclusions in the context of some non-invertible dynamical systems. For instance we treat several classes of uniformly expanding maps of the interval (for possibly unbounded functions). A general result for φ-dependent sequences is obtained in the course. (2010): 37E05, 37C30, 60F15.
Mathematics Subject Classification
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 ergodic Markov
chains.Comment: Published at http://dx.doi.org/10.1214/154957806100000202 in the
Probability Surveys (http://www.i-journals.org/ps/) by the Institute of
Mathematical Statistics (http://www.imstat.org
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.