An invariance principle for the law of the iterated logarithm for some Markov chains

Abstract

Abstract. Strassen's invariance principle for additive functionals of Markov chains with spectral gap in the Wasserstein metric is proved.

The Strassen invariance principle for certain non-stationary Markov–Feller chains

“…The results for the latter concern, for instance, positive Harris recurrent Markov chains which are assumed to be uniformly ergodic in the total variation norm (cf. [18]) and the Markov-Feller chains enjoying the exponential mixing property in the Wasserstein metric (see [1]).…”

“…Here, we also prove a version of the Strassen invariance principle for a quite general class of non-stationary Markov-Feller chains. However, on the contrary to [1], we do do not require any form of continuous dependence of the distributions of the given Markov chain on the initial conditions. A priori, we even do not demand the exponential-mixing type property (defined as in [8]).…”

“…The motivation to establish such a result derives from our research on certain random dynamical systems, developed mainly in molecular biology (see eg. the models for gene expression investigated in [11,3,17] or the model for cell cycle discussed in [16,22]), to which we were not able to apply [1,Theorem 1] directly.…”

“…Some proof techniques are adapted from [1,12], which both pertain to the martingale results by C.C. Heyde and D.J.…”

Limit theorems for some Markov chains

“…The proof of the LIL is supposed to be provided in a future paper. Some ideas useful for proving it may be adapted from Bołt et al [2]. However, we strongly believe that using an appropriate coupling measure will, again, make the proof much easier.…”