Nonconventional Random Matrix Products

TL;DR

This paper investigates the asymptotic behavior of products of nonconventional random matrices, establishing almost sure convergence of normalized log norms even with long-range dependence and nonstationarity.

Contribution

It introduces new results on the asymptotics of nonconventional random matrix products with dependent and nonstationary sequences, extending classical theory.

Findings

Normalized log norms of matrix products converge almost surely.

Results apply to sequences with long-range dependence and Markov dependence.

Provides new insights into the behavior of nonstationary random matrix products.

Abstract

Let $\xi_1,\xi_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ 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 $(X_n)$ is long-range dependent and nonstationary.