In this chapter, (continuous) positive Linnik and (nonnegative integer valued) discrete Linnik random variables are discussed. Rates of convergence and first terms of both the Edgeworth expansions and the expansions in the exponent of the distribution functions of certain sums of such random variables with nonnegative strictly stable as well as discrete stable limit laws are considered.

Positive Linnik and Discrete Linnik Distributions

The centered Poisson approximation can be considered as a link between the Poisson and normal laws. It has the structure similar to the Poisson distribution but depends on two parameters. The results obtained via the Stein–Chen method for sums of independent and dependent indicators show that the centred Poisson approximation is a strong alternative to the standard Poisson and normal approximations and, potentially, is more widely applicable.

Centered Poisson Approximation via Stein's Method

A triangular array of independent infinitesimal integer-valued random variables is considered. Asymptotic expansions for the probability distributions of sums of these variables are investigated in the case of the limiting compound Poisson laws.

Asymptotic Expansions in the Compound Poisson Limit Theorem

On a centered Poisson approximation

Large deviations in a centered Poisson approximation are examined. For lattice distributions, the centered Poisson approximation is more universal than the normal and standard Poisson laws.

V. A. Statulevičius

Papers

Asymptotic expansions in the approximation by the poisson law