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On Cramer Approximations under Violation of Cramer's Condition

Research paper by A. K. Aleskeviciene

Indexed on: 01 Oct '05Published on: 01 Oct '05Published in: Lithuanian Mathematical Journal



Abstract

Let X1, X2,... be independent identically distributed random variables with distribution function F, S0 = 0, Sn = X1 + ⋯ + Xn, and S¯n = max1⩽k⩽nSk. We obtain large-deviation theorems for Sn and S¯n under the condition 1 − F(x) = P{X1 ⩾ x} = e−l(x), l(x) = xαL(x), α ∈ (0, 1), where L(x) is a slowly varying function as x → ∞.