Compound Poisson approximation: A user's guide
type: Item
instance of: scholarly article
title: Compound Poisson approximation: A user's guide
zbMATH Open document ID: 1018.60051
author: Ourania Chryssaphinou
author: A. D. Barbour
published in: The Annals of Applied Probability
publication date: 2003-05-06T00:00:00Z
review text: This is a discussion, with many references, on approximating the distribution of \(W=\sum_\gamma X_\gamma\) by a compound Poisson distribution CP\((\lambda,{\mathbf \mu}) = \sum_{k\geq 0}\lambda^ke^{-\lambda} {\mathbf \mu}^{k*}/k!\). Here the \(X_\gamma\), \(\gamma\in \Gamma\) countable, are dependent nonnegative integer-valued random variables. Estimates are given for the total variation distance \(d_{\text{TV}}\) and the sup of the absolute difference of distribution functions. Often one has a point process consisting of clumps of weakly dependent numbers of points. Then one may assign each clump to some \(\gamma \in \Gamma\) and put \(X_\gamma = \sum_{l>0}l I_{\gamma l}\) where \(I_{\gamma l}\) is the indicator of the event that a clump of size \(l\) occurs at `time' \(\gamma\). By Stein's method for Poisson process approximation, \(d_{\text{TV}}({\mathcal L}(W),\text{CP}(\lambda,{\mathbf \mu}))\) then may be estimated. The `canonical' parameters \(\lambda\) and \(\mathbf \mu\) are \(\lambda\) = expected number of clumps and \({\mathbf \mu}(l) =\lambda^{-1}\sum_\gamma EI_{\gamma l}\). A direct approach, without clumping, is more complicated. It is accomplished by decomposing \(W\), for every \(\gamma\in \Gamma\), as \(W=W_\gamma+Z_\gamma+U_\gamma+X_\gamma\) where \(W_\gamma\) and \((X_\gamma,U_\gamma)\) should be almost independent and \(U_\gamma\), \(Z_\gamma\) not too large. Often these terms are sums of \(X_\beta\) with \(\beta\) in suitable subsets of \(\Gamma\). The canonical parameters \(\lambda\) and \(\mu\) now are sums over \(\gamma\in\Gamma\) of moments of functions of \(X_\gamma\) and \(U_\gamma\). Several estimates of the distances between \({\mathcal L}(W)\) and \(CP(\lambda,{\mathbf \mu})\) are given, often better than the above ones. Sharpening still is possible by choosing `uncanonical' \(\lambda\) and \(\mathbf\mu\). Applications: The number \(W\) of possibly overlapping \(k\)-runs of 1's in \(n\) Bernoulli experiments. The number \(W\) of possib…
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60F05
Mathematics Subject Classification ID: 60K10
Mathematics Subject Classification ID: 60J10
zbMATH DE Number: 1906223
zbMATH Keywords: compound Poisson approximation
zbMATH Keywords: applications
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aoap/1015345355
OpenAlex ID: W2084966663
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DOI: 10.1214/AOAP/1015345355
description: scientific article; zbMATH DE number 1906223