Abstract
Theorems concerning exact asymptotics of large-deviation probabilities of sums of independent random elements in a Banach space are proved. We consider probabilities of the following events: sums of independed elements belong to balls such that their centers deviate from the origin as the sample size increases. The results are a version of Bentkus' and Rachkauskas' theorems proved for the exteriors of balls centered at the origin. Bibliography: 11 titles.
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