Compound Poisson approximation: A user's guide
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 possibly overlapping increasing sequences \(X_j\), \(j= r,\dots,r+k-1\), \(k\) fixed, in a sample \(X_i\), \(i= 1,\dots,n\), from a continuous distribution. Reliability: The two-dimensional consecutive \(k\)-out-of-\(n\) system that fails if in an \(n\times n\) grid all components in a \(k\times k\) subgrid fall \((W> 0)\). The linearly arranged \(k\)-out-of-\(n\) system with \(r\) types of defects. Scan statistics: Detecting certain events in some cluster of adjacent subregions of a rectangular region in \(N^2\). The number \(W\) of copies of a given word in an i.i.d. sequence of \(n\) letters. The number \(W\) of matchings of words of length \(k\) in two such sequences, possibly in different places. The number \(W\) of visits of a stationary Markov chain to a rare set of states.
scientific article; zbMATH DE number 1906223