Nonconventional random matrix products

Abstract

Let $\xi _1,\xi _2,...$ be independent identically distributed random variables and $F:{\mathbb R}^\ell \to SL_d({\mathbb R})$ be a Borel measurable matrix-valued function. Set $X_n=F(\xi _{q_1(n)},\xi _{q_2(n)},...,\xi _{q_\ell (n)})$ where $0\leq q_1<q_2<...<q_\ell $ are increasing functions taking on integer values on integers. We study the asymptotic behavior as $N\to \infty $ of the singular values of the random matrix product $\Pi _N=X_N\cdots X_2X_1$ and show, in particular, that (under certain conditions) $\frac 1N\log \|\Pi _N\|$ converges with probability one as $N\to \infty $. We also obtain similar results for such products when $\xi _i$ form a Markov chain. The essential difference from the usual setting appears since the sequence $(X_n,\, n\geq 1)$ is long-range dependent and nonstationary.