Asymptotic expansions for distributions of compound sums of random variables with rapidly varying subexponential distribution

Content Provider	Semantic Scholar
Author	Barbe, Ph . Mccormick, W. P. Zhang, Changjin
Copyright Year	2007
Abstract	We derive an asymptotic expansion for the distribution of a compound sum of independent random variables, all having the same rapidly varying subexponential distribution. The examples of a Poisson and geometric number of summands serve as an illustration of the main result. Complete calculations are done for a Weibull distribution, with which we derive, as examples and without any difficulties, seven-term expansions. In this paper we construct asymptotic expansions for the tail area of a compound sum, when the distribution of the summands belongs to a class of rapidly varying subexponential distributions. To be more precise, let X?, i > 1, be a sequence of independent random variables, all having the same distribution, F. For any positive integer n the distribution of the partial sums Sn = X\ + + Xn is the n-fold convolution F*n. We set So = 0 and therefore F*? is defined as the distribution of the point mass at the origin. Let N be a nonnegative integer-valued random variable, independent of the X/s. We consider the distribution G of the compound sum Sn, that is E F*N, and we seek an asymptotic expansion for its tail area G = 1 ? G. First-order asymptotic results for G have been obtained by Embrechts et al. (1979), Cline (1987), and Embrechts (1985). A second-order formula may be found in Griibel (1987) and Omey and Willekens (1987). Compound sums or subordinated distributions arise as distributions of interest in several stochastic models. In insurance risk theory, it models the total claim amount. For a discussion of issues related to random sums and insurance risk, we refer the reader to Embrechts et
Starting Page	670
Ending Page	684
Page Count	15
File Format	PDF HTM / HTML
DOI	10.1239/jap/1189717537
Alternate Webpage(s)	https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6F86C96ADE629A40FBB399B7C082B965/S0021900200003351a.pdf/asymptotic_expansions_for_distributions_of_compound_sums_of_random_variables_with_rapidly_varying_subexponential_distribution.pdf
Alternate Webpage(s)	https://doi.org/10.1239/jap%2F1189717537
Volume Number	44
Language	English
Access Restriction	Open
Content Type	Text
Resource Type	Article