# Compound Poisson Approximation via Information Functionals

@article{Barbour2010CompoundPA,
title={Compound Poisson Approximation via Information Functionals},
author={Andrew D. Barbour and Oliver Johnson and Ioannis Kontoyiannis and Mokshay M. Madiman},
journal={ArXiv},
year={2010},
volume={abs/1004.3692}
}
• Published 21 April 2010
• Mathematics, Computer Science
• ArXiv
An information-theoretic development is given for the problem of compound Poisson approximation, which parallels earlier treatments for Gaussian and Poisson approximation. Nonasymptotic bounds are derived for the distance between the distribution of a sum of independent integer-valued random variables and an appropriately chosen compound Poisson law. In the case where all summands have the same conditional distribution given that they are non-zero, a bound on the relative entropy distance…

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## References

SHOWING 1-10 OF 39 REFERENCES

### On the entropy and log-concavity of compound Poisson measures

• Mathematics
ArXiv
• 2008
It is shown that the natural analog of the Poisson maximum entropy property remains valid if the measures under consideration are log-concave, but that it fails in general.

### Compound Poisson Approximation for Nonnegative Random Variables Via Stein's Method

• Mathematics
• 1992
The aim of this paper is to extend Stein's method to a compound Poisson distribution setting. The compound Poisson distributions of concern here are those of the form POIS$(\nu)$, where $\nu$ is a

### An expansion in the exponent for compound binomial approximations

• Mathematics
• 2006
The purpose of this paper is two-fold. First, we introduce a new asymptotic expansion in the exponent for the compound binomial approximation of the generalized Poisson binomial distribution. The

### Compound Poisson approximation: a user's guide

• Mathematics
• 2001
Compound Poisson approximation is a useful tool in a variety of applications, including insurance mathematics, reliability theory, and molecu- lar sequence analysis. In this paper, we review the ways

### Measure Concentration for Compound Poisson Distributions

• Mathematics
Electronic Communications in Probability
• 2006
We give a simple development of the concentration properties of compound Poisson measures on the nonnegative integers. A new modification of the Herbst argument is applied to an appropriate modified

### Accuracy of approximation in the Poisson theorem in terms of the χ2-distance

• Mathematics
• 2008
We study the limit behavior of the χ2-distance between the distributions of the nth partial sum of independent not necessarily identically distributed Bernoulli random variables and the accompanying

### Generalized Entropy Power Inequalities and Monotonicity Properties of Information

• Mathematics, Computer Science
IEEE Transactions on Information Theory
• 2007
A simple proof of the monotonicity of information in central limit theorems is obtained, both in theSetting of independent and identically distributed (i.i.d.) summands as well as in the more general setting of independent summands with variance-standardized sums.

### Stein's Method: Expository Lectures and Applications

• Mathematics
• 2004
A review of Stein’s method applied to the case of discrete random variables and attempt to complete one of Stein's open problems, that of providing a discrete version for chapter 6 of his book.

### On the Entropy of Compound Distributions on Nonnegative Integers

• Yaming Yu
• Mathematics
IEEE Transactions on Information Theory
• 2009
Two recent results of Johnson (2008) on maximum entropy characterizations of compound Poisson and compound binomial distributions are proved under fewer assumptions and with simpler arguments.

### On the Distribution of Sums of Independent Random Variables

Let {X j ; j = 1, 2,...} be a finite sequence of independent random variables. Let S = Σ X j be their sum, and let P j be the distribution of X j . Let M be the measure defined on the line deprived