On the probabilities of large deviations of the Shepp statistic

We find the asymptotic behaviour of the probability of large deviations $\mathsf P(W_{L,L}\geq\theta L)$ of the Shepp statistic $W_{L,L}$ which is equal to the maximum of fluctuations of the random walk
$$ S_n=\sum_{i=1}^n\xi_i $$
in the window of width $L$ moving in the interval $[1,2L]$ as $L\to\infty$ and $\theta$ is a constant. We assume that $\xi_1,\xi_2,\ldots$ are independent identically distributed random variables with non-lattice distribution satisfying the right-side Cramer condition. We show that the asymptotics are of the form $H_\theta L\mathsf P(S_l\geq\theta L)$, where $H_\theta$ is a constant depending on $\theta$.
This research was supported by the Russian Foundation for Basic Research, grant 01–0100–649.