Compound Poisson approximation for unbounded functions on a group, with application to large deviations
The paper establishes a second order bound on the approximation of \(\int hd \widehat Q\) by \(\int hdQ\), where \(\widehat Q = \prod^n_{i = 1} [(1 - p_i)I + p_i \mu_i]\) is the convolution of \(n\) probability measures on a measurable Abelian group \((X,B)\) which are not too far from the measure \(I\) which assigns probability 1 to the identity in \(X\), and \(Q\) is a related compound Poisson measure. The bounds are in the spirit of those in [the first author and \textit{K. P. Choi}, Ann. Probab. 20, No. 4, 1867-1876 (1992; Zbl 0764.60026)], and involve making minimal assumptions on the function \(h\), yet the bounds obtained are sharp enough to yield good large deviation estimates. Multivariate Poisson approximation is discussed as a particular application.
convolution of \(n\) probability measures
scientific article; zbMATH DE number 822730