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Estimates for interval probabilities of the sums of random variables with locally subexponential distributions

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Abstract

Let {i} i=1 be a sequence of independent identically distributed nonnegative random variables, S n = ξ1 + ⋯ +ξn. Let Δ = (0, T] and x + Δ = (x, x + T]. We study the ratios of the probabilities P(S n ε x + Δ)/P(ξ 1 ε x + Δ) for all n and x. The estimates uniform in x for these ratios are known for the so-called Δ-subexponential distributions. Here we improve these estimates for two subclasses of Δ-subexponential distributions; one of them is a generalization of the well-known class LC to the case of the interval (0, T] with an arbitrary T ≤ ∞. Also, a characterization of the class LC is given.

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Authors and Affiliations

  1. Heriot-Watt University, Edinburgh Scotland, UK

    V. V. Shneer

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  1. V. V. Shneer
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Original Russian Text Copyright c © 2006 Shneer V. V.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 47, No. 4, pp. 946–955, July–August, 2006.

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Shneer, V.V. Estimates for interval probabilities of the sums of random variables with locally subexponential distributions. Sib Math J 47, 779–786 (2006). https://doi.org/10.1007/s11202-006-0088-4

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  • DOI: https://doi.org/10.1007/s11202-006-0088-4

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