Exact Rates of Convergence for Compound Sums of Random Variables with Common Regularly Varying Distribution Function

Preprint series: 04-15 , Preprints

The paper is published: V. Antonov, C. Huber, M. Nikulin, V. Polischook (eds.) Longevity, Aging and Degradation Models in Reliability, Public Health, Medicine and Biology, Vol.2, St. Petersburg, 2004, pp. 56 - 67. ISBN 5-7422-0638-0 (Proceedings of the 1-st International French-Russian Workshop 'Longevity, Aging and Degradation Models in Reliability, Public Health, Medicine and Biology', 7-9 June 2004, St. Petersburg, Russia)

MSC:
60F05 Central limit and other weak theorems
60F10 Large deviations
60G50 Sums of independent random variables
60E07 Infinitely divisible distributions; stable distributions

Abstract: Consider a compound sum of independent and identical distributed positive random variables with common
regularly varying distribution function $F$ and an
integer-valued counting variable $\nu$ which is independent of the summands. Compound sums occur in various models in
reliability, risk theory and branching with many applications.
Consider the quotient of the distribution function of such a compound sum and the distribution function $F$
of one term.

Embrechts and Veraverbeke (1982) proved that the quotient tends to the expectation of the counting variable
as the argument $x$ tends to infinity, while Mikosch and Nagaev (2001) showed that the rate of convergence is $O(1/x)$
if $F$ satisfies some smoothness conditions.

Using non-uniform estimates with stable limit laws by Christoph and Wolf (1993), in the present paper exact rates of
convergence and the first term in asymptotic expansions are given. Examples demonstrate the results.

Keywords: Regularly varying tails, random sums, convergence rates, asymptotic expansions, stable distributions