Asymptotic expansions for generalized Poisson measures
type: Item
instance of: scholarly article
title: Asymptotic expansions for generalized Poisson measures
zbMATH Open document ID: 0931.60012
DOI: 10.1007/BF02465434
published in: Lithuanian Mathematical Journal
publication date: 1999-06-14T00:00:00Z
review text: Asymptotic expansions are obtained for distributions having the following structure: \(H = (1-p)F + pB\), where \(F \in{\mathcal F}_+\), \(B\in{\mathcal F}\), \(0 \leq p \leq c_0 < 1\), and \(\mathcal F\) (resp. \(\mathcal F_+\)) is the class of all distributions on \({\mathbb R} \) (resp. of those with nonnegative characteristic functions). Set \(\Delta_{\nu}={n\choose\nu}G^{n-\nu}(H-G)^{\nu}\) where \(G=\exp\{(H-E)-p^2(B-E)/2\}\), \(E\) is the distribution concentrated at the point \(0\), \(\nu=1,\ldots,s\) and \(s\leq [n/2]\). For instance, an upper bound is given for \(|\Delta_{\nu}|_h\) where \(h>0\) and \(|W|_h = \sup_{x}|W([x,x+h])|\) is an analogue of the Lévy concentration function for a signed measure \(W\). No moment assumptions are made. The key role belongs to the asymptotic expansion of \textit{H. Bergström} [Skand. Actuarietidskr. 1951, 1-34 (1951; Zbl 0045.07301)] and the method of triangular functions proposed by T. Arak and developed by A. Zajtsev. The results obtained can be simplified in some special cases, e.g., the author considers the distribution of a non-lattice discrete random variable. It is also mentioned that the generalized (signed) Poisson's measures used here are important for applications, especially in actuarial mathematics.
Mathematics Subject Classification ID: 60F05
zbMATH DE Number: 1296180
zbMATH Keywords: generalized Poisson measures
zbMATH Keywords: Bergström asymptotic expansion
zbMATH Keywords: Arak method
author: Vydas Čekanavičius
reviewed by: Alexander V. Bulinski
MaRDI profile type: Publication
cites work: On the rate of Poisson convergence
cites work: Q5808639
cites work: Q3806453
cites work: Approximation with accompanying distributions and asymptotic expansions. I
cites work: On asymptotic expansions in the first uniform Kolmogorov theorem
cites work: Compound Poisson approximation for unbounded functions on a group, with application to large deviations
cites work: Q4137812
cites work: Poisson approximations of multinomial distributions and point processes
cites work: Error bounds for the compound Poisson approximation
cites work: Approximation of aggregate claims distributions by compound Poisson distributions
cites work: Precision of approximation of the generalized binomial distribution by convolutions of Poisson measures
cites work: An approximation theorem for the Poisson binomial distribution
cites work: Q3485659
cites work: Approximation of Binomial Distributions by Infinitely Divisible Ones
cites work: Approximation in Variation of the Distribution of a Sum of Independent Bernoulli Variables with a Poisson Law
cites work: Approximation of sums by compound Poisson distributions with respect to stop-loss distances
cites work: Approximation of distributions of integer-valued additive functions by discrete charges. II
cites work: Q3831810
cites work: On the Accuracy of Approximation of Distributions of Sums of Independent Random Variables—Which are Nonzero with a Small Probability—By Means of Accompanying Laws
cites work: On the Uniform Approximation of Distributions of Sums of Independent Random Variables
cites work: Multidimensional Version of the Second Uniform Limit Theorem of Kolmogorov
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description: scientific article; zbMATH DE number 1296180