ИСТИНА

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 recurrent equation $Y_{n+1}= A_n Y_n + B_n$ and a random walk $S_i$ with steps $\ln A_i$. Assuming that $EA_n^{h}$ is finite for some $h>0$, $EB_n^h$, $EY_0^h$ are finite for any $h>0$ its shown that $P(Y_n\ge \exp(\theta n))\sim c P(\max_{i\le n} S_i\ge \theta n)$ some $c>0$ and any $\theta> E\ln A_n$ as $n\to\infty$. This result is applied to large deviations of branching processes in random environment.