Nonconventional random matrix products

Let ξ₁, ξ₂, … be independent identically distributed random variables and F: ℝ^l → SL_d(ℝ) be a Borel measurable matrix-valued function. Set X_n = F(ξ_q1(n), ξ_q2(n), …, ξ_ql(n)) where 0 ≤ q₁ < q₂ < … < q_l are increasing functions taking on integer values on integers. We study the asymptotic behavior as N → ∞ 1 of the singular values of the random matrix product Π_N = X_N … X₂X₁ and show, in particular, that (under certain conditions) [math] converges with probability one as N → ∞. 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, n ≥ 1) is long-range dependent and nonstationary.