Abstract

The geometric sums have been arisen from the necessity to resolve practical problems in ruin prob- ability, risk processes, queueing theory and reliability models, etc. Up to the present, the results related to geometric sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned domains of geometric attraction are still limitative. The main purpose of this article is to introduce concepts on the domain of geometric attraction of standard Laplace distribution. Using method of characteristic functions, the necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution are shown. In special case, obtained result is a weak limit theorem for geometric sums of independent and identically distributed random variables which has been well-known as the second central limit theorem. Furthermore, based on the obtained results of this paper, the analogous results for the domains of geometric attraction of exponential distribution and Linnik distribution can be established. More generally, we may extend results to the domain of geometric attraction of geometrically strictly stable distributions.
 
 

Highlights

  • University of Finance and Marketing, VietnamCopyright © VNU-HCM Press

  • Some results on the weak limit theorems for geometric sums together with rates of convergence and its applications were published by Hung (2013), Teke and Deshmukh (2014) 8,9

  • The main purpose of this paper is to show the necessary and sufficient conditions for the distribution function F which belongs to the domain of geometric attraction of standard Laplace distribution by using method of characteristic functions

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Summary

Introduction

The main purpose of this paper is to show the necessary and sufficient conditions for the distribution function F which belongs to the domain of geometric attraction of standard Laplace distribution by using method of characteristic functions. A weak limit theorem for geometric sums converging to the standard Laplace distribution is established. The necessary and sufficient conditions for a probability distribution belongs to the domain of geometric attraction of standard Laplace distribution.

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