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2 Probability and Mathematical Statistics

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

100%

Berend D. , Tassa T.

Probability and Mathematical Statistics

2010

tom 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.

A note on extremes of compound Poisson processes

80%

Berend D. , Braverman M. , Goldstein I.

tom Vol. 29, Fasc. 2

309--320

We investigate the conditions on a compound Poisson process X(t) 0≤t≤1, under which the right tail of sup0≤t≤1 X(t) is equivalent to the tail of X(1). New examples for which this relation does not hold are given.