On estimates and the asymptotic behavior of the probability of nonintersection of moving boundaries by sums of independent random variables

This paper studies estimates and the asymptotic behavior as $n\to\infty$ for the probabilities $\mathbf P\{|S_k|\leqslant f(k),\,m\leqslant k\leqslant n\}$ and $\mathbf P\{S_k\geqslant g(k), \,m\leqslant k\leqslant n\}$, where $S_n=\sum_{k=1}^n\xi_k$, the $\xi_k$ being independent identically distributed random variables with mean zero, and $f(n)$ and $g(n)$ are nonrandom functions. Under certain restrictions on the boundaries $f(n)$ and $g(n)$ logarithmic asymptotes of these probabilities are found in the case when the $\xi_k$ satisfy (respectively) a two-sided or a one-sided Cramér condition. The method is based on an absolutely continuous substitution for the original probability measure.
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