An almost sure invariance principle for some classes of inhomogeneous Markov chains and non-stationary $ρ$-mixing sequences

Yeor Hafouta

Published 2020-05-06Version 1

We prove a vector-valued almost sure invariance principle for partial sums generated by uniformly contracting or elliptic Markov chains and a uniformly bounded sequence of functions. In the real-valued case we will also consider other types of non-stationary $\rho$-mixing sequences. In the scalar case, when the variance $\sig_n^2$ of the underlying partial sums $S_n$ grows at least as fast as $n^\ve$ (for some $\ve>0$), we obtain the rate $\sig_n^{1/2+\del}$ for any $\del>0$, while in the vector-valued case, for sufficiently regular functions, we obtain the rate $s_n^{1/2+\del}$, where $s_n^2=\min_{|v|=1}v\cdot \text{Cov}(S_n)v$ is the "growth rate" of the of covariance matrix of $S_n$ in the space of positive definite matrices.