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Poisson approximation. (English) Zbl 0746.60002
Oxford Studies in Probability. 2. Oxford: Clarendon Press. x, 277 p. (1992).
This book is concentrated on the problem of Poisson approximation of sums of binomial random variables. It also considers some extensions to multivariate, compound and Poisson process approximation and deals in detail with applications to random permutations, random graphs, occupancy and urn models, spacings, exceedances and extremes. The authors of this book demonstrate the wide applicability and flexibility of the Stein-Chen method and the coupling method to obtain approximation results.
The Stein-Chen method consists in two basic steps. Firstly for subsets \(A \subset \mathbb{N},\;0<\lambda<\infty,\;P_ \lambda\) the Poisson distribution with parameter \(\lambda\), solve the equation: \[ \lambda g(j+1) - jg(j) = I(j\in A) - P_ \lambda (A),\quad j\geq 0\text{ in } g=g_{\lambda ,A}. \] As consequence one obtains for any random variable \(W\): \[ P(W\in A) - P_ \lambda (A) = E\bigl( \lambda g(W+1) - Wg(W)\bigr) . \] The right-hand side can be bounded by \[ \min\Bigl\{ 2\sup_ j | g_{\lambda ,A}(j)|,\;\sup_ j | g_{\lambda ,A} (j+1) - g_{\lambda ,A}(j)| \Bigr\} \sum_{i=1}^ n p_ i^ 2, \] for \(W=\sum_{i=1}^ n I_ i\) the sum of independent indicator functions with \(p_ i=P(I_ i=1)\). So it remains in the second step to obtain upper bounds for the factor of \(\sum p_ i^ 2\).
The book contains a lot of new or recent results in the fields described above. At some points it also shows the limits of what is obtainable by the Stein-Chen resp. coupling methods.

60A05 Axioms; other general questions in probability
60F05 Central limit and other weak theorems
60-02 Research exposition (monographs, survey articles) pertaining to probability theory