Invariance principle for stochastic processes with short memory

In this paper we give simple sufficient conditions for linear type processes with short memory that imply the invariance principle. Various examples including projective criterion are considered as applications. In particular, we treat the weak invariance principle for partial sums of linear processes with short memory. We prove that whenever the partial sums of innovations satisfy the $L_p$--invariance principle, then so does the partial sums of its corresponding linear process.Comment: Published at http://dx.doi.org/10.1214/074921706000000734 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

On the functional central limit theorem via martingale approximation

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.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ276 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Rosenthal-type inequalities for the maximum of partial sums of stationary processes and examples

The aim of this paper is to propose new Rosenthal-type inequalities for moments of order higher than 2 of the maximum of partial sums of stationary sequences including martingales and their generalizations. As in the recent results by Peligrad et al. [Proc. Amer. Math. Soc. 135 (2007) 541-550] and Rio [J. Theoret. Probab. 22 (2009) 146-163], the estimates of the moments are expressed in terms of the norms of projections of partial sums. The proofs of the results are essentially based on a new maximal inequality generalizing the Doob maximal inequality for martingales and dyadic induction. Various applications are also provided.Comment: Published in at http://dx.doi.org/10.1214/11-AOP694 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org