Theorems of large deviations in the approximation by the compound Poisson distribution
type: Item
instance of: scholarly article
title: Theorems of large deviations in the approximation by the compound Poisson distribution
zbMATH Open document ID: 1041.60025
DOI: 10.1023/A:1025771602297
published in: Acta Applicandae Mathematicae
publication date: 2003-12-09T00:00:00Z
review text: A random variable \(Y\) is said to have the compound Poisson distribution if its characteristic function \(f(t)={E}e^{itY}\) is such that \[ \log f(t)= \lambda\sum_{m=1}^{N}\bigl(e^{itm}-1\bigr)p_m,\;\lambda>0,\;p_m\geq 0,\;m=1,\ldots,N,\;p_N>0,\;\sum_{m=1}^N p_m=1. \] \(Y\) can be expressed as the sum \(\xi_1+\cdots+\xi_\eta\) of a random number of i.i.d. random variables, where \({P}(\xi_1=m)=p_m\), \(m=1,\ldots,N\), \(\eta\) has the Poisson distribution with the parameter \(\lambda>0\). Let \(X\) be a random variable taking integer nonnegative values with \({E}X={E}Y\), \({E}X^s<\infty\) for any \(s>0\). The problem is to approximate \(X\) by \(Y\). In this paper probabilities of large deviations \({P}(X\geq x)\) are investigated using the cumulant method, after a definition of cumulants \(\widetilde {\widetilde{\Gamma}}_k(X)\), suitable for the case of the compound Poisson distribution, is given. The results of the paper extend previous investigations on large deviations in the approximation by Poisson law done by the authors.
Mathematics Subject Classification ID: 60F10
zbMATH DE Number: 2015142
zbMATH Keywords: compound Poisson distribution
zbMATH Keywords: large deviations
zbMATH Keywords: cumulants
author: A. Aleškevičienė
author: V. Statulevičius
reviewed by: Birutė Kryžienė
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2015142