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Tail asymptotics of light-tailed Weibull-like sums

Asmussen S., Hashorva E., Laub P. J., Taimre T.

Probability and Mathematical Statistics

2017

Vol. 37, Fasc. 2

235--256

We consider sums of n i.i.d. random variables with tails close to exp{−xβ} for some β > 1. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg, and Resnick (1993) are discussed from the point of view of tails rather than of densities, using a somewhat different angle, and supplemented with bounds, results on a random number N of terms, and simulation algorithms.

Sums of Independent Random Variables of the Modified Makeham Distribution

Kosznik-Biernacka S.

Computational Methods in Science and Technology

2011

Vol. 17, No. 1-2

35-40

This paper shows how the renewal process in which the sums of independent random variables of the modified Makeham distribution appear can be approximated by the surrogate distribution.

Improved bounds on bell numbers and on moments of sums of random variables

Berend D., Tassa T.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

185--205

We provide bounds for moments of sums of sequences of independent random variables. Concentrating on uniformly bounded nonnegative random variables, we are able to improve upon previous results due to Johnson et al. [10] and Latała [12]. Our basic results provide bounds involving Stirling numbers of the second kind and Bell numbers. By deriving novel effective bounds on Bell numbers and the related Bell function, we are able to translate our moment bounds to explicit ones, which are tighter than previous bounds. The study was motivated by a problem in operation research, in which it was required to estimate the Lp-moments of sums of uniformly bounded non-negative random variables (representing the processing times of jobs that were assigned to some machine) in terms of the expectation of their sum.