Abstract
Let X 1, X 2,... be independent identically distributed random variables with distribution function F, S 0 = 0, S n = X 1 + ⋯ + X n , and S¯ n = max1⩽k⩽n S k . We obtain large-deviation theorems for S n and S¯ n under the condition 1 − F(x) = P{X 1 ⩾ x} = e−l(x), l(x) = x α L(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞.
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Translated from Lietuvos Matematikos Rinkinys, Vol. 45, No. 4, pp. 447–456, October–December, 2005.
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Aleskeviciene, A.K. On Cramer Approximations under Violation of Cramer's Condition. Lith Math J 45, 359–367 (2005). https://doi.org/10.1007/s10986-006-0001-7
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DOI: https://doi.org/10.1007/s10986-006-0001-7