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Nonconventional Random Matrix Products
Yuri Kifer, Sasha Sodin · 2018-03-25 · via math.PR updates on arXiv.org

Let $ξ_1,ξ_2,...$ be independent identically distributed random variables and $F:\bbR^\ell\to SL_d(\bbR)$ be a Borel measurable matrix-valued function. Set $X_n=F(ξ_{q_1(n)},ξ_{q_2(n)},...,ξ_{q_\ell(n)})$ 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 $Π_N=X_N\cdots X_2X_1$ and show, in particular, that (under certain conditions) $\frac 1N\log\|Π_N\|$ converges with probability one as $N\to\infty$. We also obtain similar results for such products when $ξ_i$ form a Markov chain. The essential difference from the usual setting appears since the sequence $(X_n)$ is long-range dependent and nonstationary.

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