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The rate of convergence for subexponential distributions

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Abstract

A distribution functionF on the nonnegative real line is called subexponential if

$$\mathop {\lim }\limits_{x \to \infty } \left( {1 - F^{*n} (x)} \right)/\left( {1 - F(x)} \right) = n, for all n \geqslant 2,$$

whereF *n denotes then-fold Stieltjes convolution ofF with itself. In this paper, we consider the rate of convergence in the above definition and we discuss the asymptotic behavior ofR n (x) defined byR n (x)=1−F *n (x)−n(1−F(x)). Our results complement those previously obtained by several authors. In this paper, we define several new classes of functions related to regular variation andO-regular variation. As a typical result, in one of our theorems we show thatR n (x)=O(1)f(x)R(x), wheref(x) is the density ofF andR(x)=∫ x0 (1−F(y))dy. We also discuss some applications.

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Authors and Affiliations

  1. Institute of Mathematics and Informatics, Akademijos 4, 2600, Vilnius, Lithuania

    A. Baltrūnas

  2. EHSAL, Stormstraat 2, 1000, Brussel, Belgium

    E. Omey

Authors
  1. A. Baltrūnas
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  2. E. Omey
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Additional information

Published in Lietuvos Matematikos Rinkinys, Vol. 38, No. 1, pp. 1–18, January–March, 1998. Original article submitted April 24, 1996.

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Baltrūnas, A., Omey, E. The rate of convergence for subexponential distributions. Lith Math J 38, 1–14 (1998). https://doi.org/10.1007/BF02465540

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