Abstract
This paper studies estimates and the asymptotic behavior as for the probabilities , and , , where , the being independent identically distributed random variables with mean zero, and and are nonrandom functions. Under certain restrictions on the boundaries and logarithmic asymptotes of these probabilities are found in the case when the satisfy (respectively) a two-sided or a one-sided Cramer condition. The method is based on an absolutely continuous substitution for the original probability measure. Bibliography: 18 titles.
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