New uniform asymptotic approximations with error bounds are derived for a generalized total variation distance of Poisson approximations to the Poisson-binomial distribution. The method of proof is also applicable to other Poisson approximation problems. MSC 2000 Subject Classifications: Primary 62E17; secondary 60C05.

The distributions of the run occurrences for a sequence of independent and identically distributed (i.i.d.) experiments are usually obtained by combinatorial methods (see Balakrishnan and Koutras (2002, Chapter 5)) and the resulting formulae are often very tedious, while the distributions for non i.i.d. experiments are generally intractable. It is therefore of practical interest to find a suita...

In this paper, we discuss the distributional properties of random sums. We first derive conditions under which the distribution of a binomial sum is PF2 and then show under the same conditions the distribution of a Poisson sum is PF2 by approximating a Poisson sum by a sequence of binomial sums. The PF2 property reveals the monotonicity property of the reversed failure rates of certain compound...

We apply the Stein-Chen method for Poisson approximation to spin-half Isingtype models, in positive external eld, which satisfy the FKG inequality. In particular we show that, provided the density of minus spins is low and can be expanded as a convergent power series in the activity (fugacity) variable, the distribution of minus spins is approximately Poisson. The error of the approximation is ...

A d-dimensional ferromagnetic Ising model on a lattice torus is considered. As the size of the lattice tends to infinity, two conditions ensuring a Poisson approximation for the distribution of the number of occurrences in the lattice of any given local configuration are suggested. The proof builds on the Stein-Chen method. The rate of the Poisson approximation and the speed of convergence to i...

Poisson approximation has received extensive interest in recent probability literature. The general problem is to derive bounds on the closeness of a suitable Poisson distribution to the distribution in question (usually of discrete type). To measure the degree of proximity of Poisson approximation, the following distances are usually introduced: for any two random variables X and Y taking valu...

In this paper we study approximating the total loss associated with the individual insurance risk model by a compound Poisson random variable. By minimizing the expectation of the absolute deviation of the compound Poisson random variable from the true total loss, we investigate not only the optimal compound Poisson random variable but also the numerical calculation of the approximation error. ...