  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.
