Abstract
The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric.
 Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.
Highlights
We follow the notations used in 1
It is seen that when r = 1, the negative-binomial sum reduces to a geometric sum
Using Gnedenko’s Transfer Theorem, a weak limit theorem for negative-binomial sum of i.i.d. random variables will be {established}as follows Theorem 2
Summary
We follow the notations used in 1. A random variable Nr,p is said to have negative-binomial distribution with two parameters p ∈ (0, 1) andr ∈ N, if its probability mass function is given in form Using Zolotarev’s probability metric, the rate of convergence in weak limit theorem for normalized negative-binomial sum (pn/r)1/α SNr,pn will be estimated.
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