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# Colnpound Poisson Approximation for Sums of Depe12dent Raadorn Variables

DOI link for Colnpound Poisson Approximation for Sums of Depe12dent Raadorn Variables

Colnpound Poisson Approximation for Sums of Depe12dent Raadorn Variables book

# Colnpound Poisson Approximation for Sums of Depe12dent Raadorn Variables

DOI link for Colnpound Poisson Approximation for Sums of Depe12dent Raadorn Variables

Colnpound Poisson Approximation for Sums of Depe12dent Raadorn Variables book

## ABSTRACT

Let X I , X g , ..., X,, be a sequence of real-valucd, nonnegative r.v.s. The subject of the present paper is the approxirnation o f the distribution of the s u n of X, when the niasses of C(X,), 1; = 1 , 2 , . .. , n are concentrated around 0; and X,, i - 1 , 2 , ..., 12 are "weakly" dependent. S1lcl.1 problems arise quite naturally w1m1 wc are tfealing with a set of "rare" and "itlmost uiirelated" events. The simplest cssc when Xi arc binary i.i.tl. r.v.s was initially treated in the funtlarnental work of Poisson as early as 1837 when he establislicd the c1assic;~l Poissou approxintt~tion of the binomial disl;ribution. Sir~ce then a huge variety of generalizatior~s ancl extensions of this result has appeared in the literature. In sonic of therii the assunipt io t~ of identical X , was rclaxed while ill others the assumption of intlependence was replaced by i i condition of "local" or "weak deperdence". The latter generalization was thoroughly investigated after the introduction of the rnucli acclaimed Stein-Chcn method (c.f. Chcn (1975); the

Section 4.4 two specific exarr~ples are considered: the first one cieals with the distribution of the total exceedance amount (above a prespecified threshold) of moving snms in a sequence of i.i.d. r.v.s. and the second with the distribution of the number of overlapping success runs in a sequence of Markov dependent trials. In both cases: a bound for the distance between the distribution of interest and a proper corr~pound Poisson distribution is obtained arid an asymptotic result (compound Poisson convergence) is established.