Large Deviations for Solution of Random Recurrence Equation

#### A. V. Shklyaev

2016, v.22, №1, 139-164

ABSTRACT

Let $(A_n,B_n)$ be a sequence of i.i.d.\ random vectors with $A_n>0$, $B_n>0$ a.s. Consider a random recurrence equation $Y_{n+1}= A_n Y_n + B_n$ and a random walk $S_i$ with steps $\ln A_i$. Assuming that $EA_n^{h}, EB_n^h, EY_0^h$ are finite for some $h>0$ we show that $P(Y_n\ge \exp(\theta n))\sim c P(\max_{i\le n} S_i\ge \theta n)$ for some $c>0$.

Keywords: large deviations, random walks, random equations, random recurrence equations