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Poisson approximation to the convolution of PSDs

Kumar A. N., Vellaisamy P., Viens F.

Probability and Mathematical Statistics

2022

Vol. 42, Fasc. 1

63--80

In this article, we obtain, for the total variation distance, error bounds for Poisson approximation to the convolution of power series distributions via Stein's method. This provides a unified approach to many known discrete distributions. Several Poisson limit theorems follow as corollaries from our bounds. As applications, we compare Poisson approximation results with negative binomial approximation results for sums of Bernoulli, geometric, and logarithmic series random variables.

Fractional negative binomial and Pólya processes

Vellaisamy P., Maheshwari A.

Probability and Mathematical Statistics

2018

Vol. 38, Fasc. 1

77--101

In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process (SFPP) is defined by replacing the rate parameter λ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to PDEs governing the density of the FNBP and the SFPP are also investigated.

Compound negative binomial approximations for sums of random variables

Vellaisamy P., Upadhye S.

Probability and Mathematical Statistics

2009

Vol. 29, Fasc. 2

205--226

The negative binomial approximations arise in telecommunications, network analysis and population genetics, while compound negative binomial approximations arise, for example, in insurance mathematics. In this paper, we first discuss the approximation of the sum of independent, but not identically distributed, geometric (negative binomial) random variables by a negative binomial distribution, using Kerstan’s method and the method of exponents. The appropriate choices of the parameters of the approximating distributions are also suggested. The rates of convergence obtained here improve upon, under certain conditions, some of the known results in the literature. The related Poisson convergence result is also studied. We then extend Kerstan’s method to the case of compound negative binomial approximations and error bounds for the total variation metric are obtained. The approximation by a suitable finite signed measure is also studied. Some interesting special cases are investigated in detail and a few examples are discussed as well.