Negative binomial sums of random variables and discounted reward processes

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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Finiteness of Record values and Alternative Asymptotic Theory of Records with Atom Endpoints

Afrika Statistika ◽

10.16929/as/2020.2371.163 ◽

2020 ◽

Vol 15 (3) ◽

pp. 2371-2385

Author(s):

Gane Samb Lo ◽

Harouna Sangaré ◽

Cherif Mamadou Moctar Traoré ◽

Mohammad Ahsanullah

Keyword(s):

Negative Binomial ◽

Sufficient Conditions ◽

Random Variables ◽

Random Variable ◽

Record Values ◽

Cumulative Distribution ◽

Hitting Times ◽

Record Times ◽

Binomial Random Variable ◽

Upper Record Values

Asymptotic theories on record values and times, including central limit theorems, make sense only if the sequence of records values (and of record times) is infinite. If not, such theories could not even be an option. In this paper, we give necessary and/or sufficient conditions for the finiteness of the number of records. We prove, for example for iid real valued random variable, that strong upper record values are finite if and only if the upper endpoint is finite and is an atom of the common cumulative distribution function. The only asymptotic study left to us concerns the infinite sequence of hitting times of that upper endpoints, which by the way, is the sequence of weak record times. The asymptotic characterizations are made using negative binomial random variables and the dimensional multinomial random variables. Asymptotic comparison in terms of consistency bounds and confidence intervals on the different sequences of hitting times are provided. The example of a binomial random variable is given.

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On the Order of Approximation in Limit Theorems for Negative–Binomial Sums of Strictly Stationary m-Dependent Random Variables

Acta Mathematica Vietnamica ◽

10.1007/s40306-020-00406-x ◽

2021 ◽

Author(s):

Tran Loc Hung ◽

Phan Tri Kien

Keyword(s):

Limit Theorems ◽

Negative Binomial ◽

Random Variables ◽

Order Of Approximation ◽

Dependent Random Variables ◽

Binomial Sums

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

Science and Technology Development Journal ◽

10.32508/stdj.v22i4.1689 ◽

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.

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A refinement of normal approximation to Poisson binomial

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/ijmms.2005.717 ◽

2005 ◽

Vol 2005 (5) ◽

pp. 717-728 ◽

Cited By ~ 4

Author(s):

K. Neammanee

Keyword(s):

Normal Distribution ◽

Normal Approximation ◽

Random Variables ◽

Random Variable ◽

Standard Normal Distribution ◽

Bernoulli Random Variables ◽

Binomial Random Variable ◽

Standard Normal ◽

Correction Terms ◽

Better Than

LetX1,X2,…,Xnbe independent Bernoulli random variables withP(Xj=1)=1−P(Xj=0)=pjand letSn:=X1+X2+⋯+Xn.Snis called a Poisson binomial random variable and it is well known that the distribution of a Poisson binomial random variable can be approximated by the standard normal distribution. In this paper, we use Taylor's formula to improve the approximation by adding some correction terms. Our result is better than before and is of order1/nin the casep1=p2=⋯=pn.

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Poisson and negative binomial item count techniques for surveys with sensitive question

Statistical Methods in Medical Research ◽

10.1177/0962280214563345 ◽

2014 ◽

Vol 26 (2) ◽

pp. 931-947 ◽

Cited By ~ 7

Author(s):

Guo-Liang Tian ◽

Man-Lai Tang ◽

Qin Wu ◽

Yin Liu

Keyword(s):

Confidence Interval ◽

Negative Binomial ◽

Survey Design ◽

Random Variable ◽

Parameter Estimate ◽

Original Design ◽

Bernoulli Random Variables ◽

Binomial Random Variable ◽

Original Item ◽

Item Count Technique

Although the item count technique is useful in surveys with sensitive questions, privacy of those respondents who possess the sensitive characteristic of interest may not be well protected due to a defect in its original design. In this article, we propose two new survey designs (namely the Poisson item count technique and negative binomial item count technique) which replace several independent Bernoulli random variables required by the original item count technique with a single Poisson or negative binomial random variable, respectively. The proposed models not only provide closed form variance estimate and confidence interval within [0, 1] for the sensitive proportion, but also simplify the survey design of the original item count technique. Most importantly, the new designs do not leak respondents’ privacy. Empirical results show that the proposed techniques perform satisfactorily in the sense that it yields accurate parameter estimate and confidence interval.

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Sums of random variables having the modified geometric distribution by application to two-person games

Advances in Applied Probability ◽

10.2307/1426639 ◽

1978 ◽

Vol 10 (3) ◽

pp. 647-665 ◽

Cited By ~ 5

Author(s):

M. J. Phillips

Keyword(s):

Negative Binomial Distribution ◽

Binomial Distribution ◽

Negative Binomial ◽

Geometric Distribution ◽

Random Variables ◽

Sums Of Random Variables ◽

Pascal Distribution ◽

Person Game ◽

Independent And Identically Distributed

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.

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Compound Poisson point processes, concentration and oracle inequalities

Journal of Inequalities and Applications ◽

10.1186/s13660-019-2263-8 ◽

2019 ◽

Vol 2019 (1) ◽

Cited By ~ 2

Author(s):

Huiming Zhang ◽

Xiaoxu Wu

Keyword(s):

Point Process ◽

Poisson Point Process ◽

Point Processes ◽

Negative Binomial ◽

Random Variable ◽

Acta Math ◽

Concentration Inequalities ◽

Oracle Inequalities ◽

Compound Poisson ◽

Binomial Random Variable

AbstractThis note aims at presenting several new theoretical results for the compound Poisson point process, which follows the work of Zhang et al. (Insur. Math. Econ. 59:325–336, 2014). The first part provides a new characterization for a discrete compound Poisson point process (proposed by Aczél (Acta Math. Hung. 3(3):219–224, 1952)), it extends the characterization of the Poisson point process given by Copeland and Regan (Ann. Math. 37:357–362, 1936). Next, we derive some concentration inequalities for discrete compound Poisson point process (negative binomial random variable with unknown dispersion is a significant example). These concentration inequalities are potentially useful in count data regression. We give an application in the weighted Lasso penalized negative binomial regressions whose KKT conditions of penalized likelihood hold with high probability and then we derive non-asymptotic oracle inequalities for a weighted Lasso estimator.

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Sums of random variables having the modified geometric distribution by application to two-person games

Advances in Applied Probability ◽

10.1017/s0001867800031098 ◽

1978 ◽

Vol 10 (03) ◽

pp. 647-665

Author(s):

M. J. Phillips

Keyword(s):

Negative Binomial Distribution ◽

Binomial Distribution ◽

Negative Binomial ◽

Geometric Distribution ◽

Random Variables ◽

Sums Of Random Variables ◽

Pascal Distribution ◽

Person Game ◽

Independent And Identically Distributed

The distribution of a fixed sum of independent and identically distributed random variables with the modified geometric distribution is the same as the distribution obtained by the compounding by a binomial distribution of either a negative binomial distribution or a Pascal distribution. This result can be used to obtain three summations for the game score probabilities of a two-person game, and leads to the consideration of various ways of dividing up the trials of the game. The game score probabilities are then used to consider the ‘fairness’ of four games and to analyse various methods of ‘setting’ (or ‘tie-breaking’) the games.

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COMPOUND RANDOM VARIABLES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964804184039 ◽

2004 ◽

Vol 18 (4) ◽

pp. 473-484 ◽

Cited By ~ 4

Author(s):

Erol Peköz ◽

Sheldon M. Ross

Keyword(s):

Arbitrary Function ◽

Negative Binomial ◽

Random Variables ◽

Random Variable ◽

Mass Function ◽

Probability Mass Function ◽

Probabilistic Proof ◽

Recursive Formulas ◽

Probability Mass ◽

Positive Compound

We give a probabilistic proof of an identity concerning the expectation of an arbitrary function of a compound random variable and then use this identity to obtain recursive formulas for the probability mass function of compound random variables when the compounding distribution is Poisson, binomial, negative binomial random, hypergeometric, logarithmic, or negative hypergeometric. We then show how to use simulation to efficiently estimate both the probability that a positive compound random variable is greater than a specified constant and the expected amount by which it exceeds that constant.

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