scholarly journals Asymptotic behaviors with convergence rates of distributions of negative-binomial sums

2020 ◽  
Vol 22 (4) ◽  
pp. 415-421
Author(s):  
Tran Loc Hung ◽  
Phan Tri Kien ◽  
Nguyen Tan Nhut
Keyword(s):  
Limit Theorems ◽  
Negative Binomial ◽  
Convergence Rates ◽  
Random Variables ◽  
Rates Of Convergence ◽  
Weak Limit ◽  
Probability Metric ◽  
Weak Limit Theorems ◽  
Distribution Theorem ◽  
Binomial Sums

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.

scholarly journals On chi-square type distributions with geometric degrees of freedom in relation to geometric sums

2019 ◽  
Vol 22 (1) ◽  
pp. 180-184
Author(s):  
Tran Loc Hung
Keyword(s):  
Degrees Of Freedom ◽  
Limit Theorems ◽  
Random Variables ◽  
Weak Limit ◽  
Random Sums ◽  
Chi Square ◽  
Asymptotic Behaviors ◽  
Weak Limit Theorems ◽  
Geometric Sums ◽  
Applied Fields

The chi-square distribution with degrees of freedom has an important role in probability, statistics and various applied fields as a special probability distribution. This paper concerns the relations between geometric random sums and chi-square type distributions whose degrees of freedom are geometric random variables. Some characterizations of chi-square type random variables with geometric degrees of freedom are calculated. Moreover, several weak limit theorems for the sequences of chi-square type random variables with geometric random degrees of freedom are established via asymptotic behaviors of normalized geometric random sums.

A study of chi-square-type distribution with geometrically distributed degrees of freedom in relation to distributions of geometric random sums

2020 ◽  
Vol 70 (1) ◽  
pp. 213-232
Author(s):  
Tran Loc Hung
Keyword(s):  
Degrees Of Freedom ◽  
Limit Theorems ◽  
Random Variables ◽  
Weak Limit ◽  
Sums Of Squares ◽  
Random Sums ◽  
Chi Square ◽  
Type Distribution ◽  
Standard Normal ◽  
Weak Limit Theorems

AbstractThe purpose of this paper is to study a chi-square-type distribution who degrees of freedom are geometric random variables in connection with weak limiting distributions of geometric random sums of squares of independent, standard normal distributed random variables. Some characteristics of chi-square-type random variables with geometrically distributed degrees of freedom including probability density function, probability distribution function, mean and variance are calculated. Some asymptotic behaviors of chi-square-type random variables with geometrically distributed degrees of freedom are also established via weak limit theorems for normalized geometric random sums of squares of independent, standard normal distributed random variables. The rates of convergence in desired weak limit theorems also estimated through Trotter’s distance. The received results are extensions and generalizations of several known results.

Convergence rates in the law of large numbers. II

1968 ◽  
Vol 64 (2) ◽  
pp. 485-488 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Convergence Rates ◽  
Random Variables ◽  
Almost Sure Convergence ◽  
Random Variable ◽  
Rates Of Convergence ◽  
Independent Random Variables ◽  
Large Numbers ◽  
Present Context ◽  
Image Position ◽  
Uniformly Bounded

Let {Xn: n ≥ 1} be a sequence of independent random variables and write Suppose that the random vairables Xn are uniformly bounded by a random variable X in the sense thatSet qn(x) = Pr(|Xn| > x) and q(x) = Pr(|Xn| > x). If qn ≤ q and E|X|r < ∞ with 0 < r < 2 then we have (see Loève(4), 242)where ak = 0, if 0 < r < 1, and = EXk if 1 ≤ r < 2 and ‘a.s.’ stands for almost sure convergence. the purpose of this paper is to study the rates of convergence ofto zero for arbitrary ε > 0. We shall extend to the present context, results of (3) where the case of identically distributed random variables was treated. The techniques used here are strongly related to those of (3).

Negative binomial sums of random variables and discounted reward processes

1998 ◽  
Vol 35 (3) ◽  
pp. 589-599
Author(s):  
William L. Cooper
Keyword(s):  
Differential Equation ◽  
Negative Binomial ◽  
Infinite Horizon ◽  
Random Variables ◽  
Random Variable ◽  
Random Time ◽  
Sums Of Random Variables ◽  
Total Reward ◽  
Binomial Random Variable ◽  
Binomial Sums

Given a sequence of random variables (rewards), the Haviv–Puterman differential equation relates the expected infinite-horizon λ-discounted reward and the expected total reward up to a random time that is determined by an independent negative binomial random variable with parameters 2 and λ. This paper provides an interpretation of this proven, but previously unexplained, result. Furthermore, the interpretation is formalized into a new proof, which then yields new results for the general case where the rewards are accumulated up to a time determined by an independent negative binomial random variable with parameters k and λ.

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