Exponential Growth of Products of Non-Stationary Markov-Dependent Matrices

Abstract Let $(\xi _j)_{j\ge 1} $ be a non-stationary Markov chain with phase space $X$ and let $\mathfrak {g}_j:\,X\mapsto \textrm {SL}(m,{\mathbb {R}})$ sequence of functions on values in the unimodular group. Set $g_j=\mathfrak {g}_j(\xi _j)$ denote by $S_n=g_n\ldots g_1$, product matrices $g_j$. We provide sufficient conditions for exponential growth norm $\|S_n\|$ when is not supposed to stationary. This generalizes classical theorem Furstenberg products independent identically distributed as well its extension Virtser stationary Markov-dependent matrices.