We derive a new maximal inequality for stationary sequences under a martingale-type condition introduced by Maxwell and Woodroofe [Ann. Probab. 28 (2000) 713-724]. Then, we apply it to establish the Donsker invariance principle for this class of stationary sequences. A Markov chain example is given in order to show the optimality of the conditions imposed.
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
We establish the central limit theorem for linear processes with dependent
innovations including martingales and mixingale type of assumptions as defined
in McLeish [Ann. Probab. 5 (1977) 616--621] and motivated by Gordin [Soviet
Math. Dokl. 10 (1969) 1174--1176]. In doing so we shall preserve the generality
of the coefficients, including the long range dependence case, and we shall
express the variance of partial sums in a form easy to apply. Ergodicity is not
required.Comment: Published at http://dx.doi.org/10.1214/009117906000000179 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Abstract. The paper aims to establish a new sharp Burkholder-type maximal inequality in L p for a class of stationary sequences that includes martingale sequences, mixingales and other dependent structures. The case when the variables are bounded is also addressed, leading to an exponential inequality for a maximum of partial sums. As an application we present an invariance principle for partial sums of certain maps of Bernoulli shifts processes.
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