NWU property of a class of random sums

2000 ◽  
Vol 37 (1) ◽  
pp. 283-289 ◽  
Author(s):  
Jun Cai ◽  
Vladimir Kalashnikov
Keyword(s):  
Renewal Process ◽  
Negative Binomial ◽  
Random Sums ◽  
Poisson Distributions ◽  
Geometric Sums ◽  
Binomial Sums

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

NWU property of a class of random sums

2000 ◽  
Vol 37 (01) ◽  
pp. 283-289 ◽  
Author(s):  
Jun Cai ◽  
Vladimir Kalashnikov
Keyword(s):  
Renewal Process ◽  
Negative Binomial ◽  
Random Sums ◽  
Poisson Distributions ◽  
Geometric Sums ◽  
Binomial Sums

In this note, we derive an inequality for the renewal process. Then, using this inequality, together with an identity in terms of the renewal process for the tails of random sums, we prove that a class of random sums is always new worse than used (NWU). Thus, the well-known NWU property of geometric sums is extended to the class of random sums. This class is illustrated by some examples, including geometric sums, mixed geometric sums, certain mixed Poisson distributions and certain negative binomial sums.

scholarly journals On the Accuracy of the Generalized Gamma Approximation to Generalized Negative Binomial Random Sums

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1571
Author(s):  
Irina Shevtsova ◽  
Mikhail Tselishchev
Keyword(s):  
Negative Binomial Distribution ◽  
Binomial Distribution ◽  
Law Of Large Numbers ◽  
Negative Binomial ◽  
Infinite Divisibility ◽  
Random Sums ◽  
Generalized Gamma ◽  
Second Moments ◽  
Large Numbers ◽  
Generalized Negative Binomial Distribution

We investigate the proximity in terms of zeta-structured metrics of generalized negative binomial random sums to generalized gamma distribution with the corresponding parameters, extending thus the zeta-structured estimates of the rate of convergence in the Rényi theorem. In particular, we derive upper bounds for the Kantorovich and the Kolmogorov metrics in the law of large numbers for negative binomial random sums of i.i.d. random variables with nonzero first moments and finite second moments. Our method is based on the representation of the generalized negative binomial distribution with the shape and exponent power parameters no greater than one as a mixed geometric law and the infinite divisibility of the negative binomial distribution.

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.

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.

scholarly journals Mixed Compound Poisson Distributions

1986 ◽  
Vol 16 (S1) ◽  
pp. S59-S79 ◽  
Author(s):  
Gord Willmot
Keyword(s):  
Negative Binomial ◽  
Random Variable ◽  
Computational Formula ◽  
Inverse Gaussian ◽  
Poisson Distributions ◽  
Mixing Distributions ◽  
Chi Squared ◽  
General Distributions ◽  
Generalized Inverse Gaussian ◽  
Thick Tails

AbstractThe distribution of total claims payable by an insurer is considered when the frequency of claims is a mixed Poisson random variable. It is shown how in many cases the total claims density can be evaluated numerically using simple recursive formulae (discrete or continuous).Mixed Poisson distributions often have desirable properties for modelling claim frequencies. For example, they often have thick tails which make them useful for long-tailed data. Also, they may be interpreted as having arisen from a stochastic process. Mixing distributions considered include the inverse Gaussian, beta, uniform, non-central chi-squared, and the generalized inverse Gaussian as well as other more general distributions.It is also shown how these results may be used to derive computational formulae for the total claims density when the frequency distribution is either from the Neyman class of contagious distributions, or a class of negative binomial mixtures. Also, a computational formula is derived for the probability distribution of the number in the system for the M/G/1 queue with bulk arrivals.

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.

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