Convergence of lower records and infinite divisibility

2003 ◽  
Vol 40 (04) ◽  
pp. 865-880 ◽  
Author(s):  
Arup Bose ◽  
Sreela Gangopadhyay ◽  
Anish Sarkar ◽  
Arindam Sengupta
Keyword(s):  
Random Variable ◽  
Infinite Divisibility ◽  
Partial Sums ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Infinite Sums ◽  
Necessary And Sufficient ◽  
Lower Records

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

Convergence of lower records and infinite divisibility

2003 ◽  
Vol 40 (4) ◽  
pp. 865-880 ◽  
Author(s):  
Arup Bose ◽  
Sreela Gangopadhyay ◽  
Anish Sarkar ◽  
Arindam Sengupta
Keyword(s):  
Random Variable ◽  
Infinite Divisibility ◽  
Partial Sums ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
The Laplace Transform ◽  
Infinite Sums ◽  
Necessary And Sufficient ◽  
Lower Records

We study the properties of sums of lower records from a distribution on [0,∞) which is either continuous, except possibly at the origin, or has support contained in the set of nonnegative integers. We find a necessary and sufficient condition for the partial sums of lower records to converge almost surely to a proper random variable. An explicit formula for the Laplace transform of the limit is derived. This limit is infinitely divisible and we show that all infinitely divisible random variables with continuous Lévy measure on [0,∞) originate as infinite sums of lower records.

scholarly journals A limit theorem for the tails of discrete infinitely divisible laws with applications to fluctuation theory

1982 ◽  
Vol 32 (3) ◽  
pp. 412-422 ◽  
Author(s):  
Paul Embrechts ◽  
John Hawkes
Keyword(s):  
Domain Of Attraction ◽  
Divisible Distribution ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Stable Law ◽  
Infinitely Divisible Laws ◽  
Ladder Epoch ◽  
Necessary And Sufficient

AbstractSuppose that (pn) is an infinitely divisible distribution on the non-negative integers having Lévy measure (vn). In this paper we derive a necessary and sufficient condition for the existence of the limit limn→∞ pn/vn. We also derive some other results on the asymptotic behaviour of the sequence (Pn) and apply some of our results to the theory of fluctuations of random walks. We obtain a necessary and sufficient condition for the first positive ladder epoch to belong to the domain of attraction of a spectrally positive stable law with index α, α ∈ (1,2).

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

A characterization of the exponential distribution

1972 ◽  
Vol 9 (02) ◽  
pp. 457-461 ◽  
Author(s):  
M. Ahsanullah ◽  
M. Rahman
Keyword(s):  
Probability Distribution ◽  
Order Statistics ◽  
Lebesgue Measure ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Positive Random Variable ◽  
Absolutely Continuous ◽  
Necessary And Sufficient

A necessary and sufficient condition based on order statistics that a positive random variable having an absolutely continuous probability distribution (with respect to Lebesgue measure) will be exponential is given.

On the best-choice problem when the number of observations is random

1983 ◽  
Vol 20 (1) ◽  
pp. 165-171 ◽  
Author(s):  
Joseph D. Petruccelli
Keyword(s):  
Optimal Stopping ◽  
Random Variable ◽  
Stopping Rules ◽  
Necessary And Sufficient Condition ◽  
Choice Problem ◽  
Best Choice Problem ◽  
Optimal Stopping Rule ◽  
Optimal Stopping Rules ◽  
Bounded Random Variable ◽  
Necessary And Sufficient

We consider the problem of maximizing the probability of choosing the largest from a sequence of N observations when N is a bounded random variable. The present paper gives a necessary and sufficient condition, based on the distribution of N, for the optimal stopping rule to have a particularly simple form: what Rasmussen and Robbins (1975) call an s(r) rule. A second result indicates that optimal stopping rules for this problem can, with one restriction, take virtually any form.

Max-infinite divisibility

1977 ◽  
Vol 14 (02) ◽  
pp. 309-319 ◽  
Author(s):  
A. A. Balkema ◽  
S. I. Resnick
Keyword(s):  
Distribution Function ◽  
Sufficient Conditions ◽  
Infinite Divisibility ◽  
Necessary And Sufficient Conditions ◽  
Infinitely Divisible ◽  
Extremal Processes ◽  
Necessary And Sufficient

Necessary and sufficient conditions are given for a distribution function in ℝ2 to be max-infinitely divisible. The d.f. F is max i.d. if F t is a d.f. for every t > 0. This property is essential in defining multivariate extremal processes and arises in an approach to the study of the range of an i.i.d. sample.

scholarly journals On characterizations and infinite divisibility of recently introduced distributions

2016 ◽  
Vol 53 (4) ◽  
pp. 467-511 ◽  
Author(s):  
G. G. Hamedani
Keyword(s):  
Weak Convergence ◽  
Order Statistic ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Simple Relationship ◽  
Infinitely Divisible

We present here characterizations of the most recently introduced continuous univariate distributions based on: (i) a simple relationship between two truncated moments; (ii) truncated moments of certain functions of the 1th order statistic; (iii) truncated moments of certain functions of the nth order statistic; (iv) truncated moment of certain function of the random variable. We like to mention that the characterization (i) which is expressed in terms of the ratio of truncated moments is stable in the sense of weak convergence. We will also point out that some of these distributions are infinitely divisible via Bondesson’s 1979 classifications.

Convolution equivalence and infinite divisibility

2004 ◽  
Vol 41 (02) ◽  
pp. 407-424 ◽  
Author(s):  
Anthony G. Pakes
Keyword(s):  
Distribution Function ◽  
Poisson Distribution ◽  
Infinite Divisibility ◽  
Compound Poisson Distribution ◽  
Infinitely Divisible Distributions ◽  
Random Sum ◽  
Lévy Measure ◽  
Infinitely Divisible ◽  
Tail Behaviour ◽  
Equivalence Result

Known results relating the tail behaviour of a compound Poisson distribution function to that of its Lévy measure when one of them is convolution equivalent are extended to general infinitely divisible distributions. A tail equivalence result is obtained for random sum distributions in which the summands have a two-sided distribution.

Some Reliability and Other Properties of Beta-Transformed Random Variables

2018 ◽  
Vol 33 (2) ◽  
pp. 83-92
Author(s):  
M. Sreehari ◽  
E. Sandhya ◽  
V. K. Mohamed Akbar
Keyword(s):  
Power Series ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Power Series Distribution ◽  
Geometric Random Variable ◽  
Necessary And Sufficient

Abstract The reliability properties of beta-transformed random variables are discussed. A necessary and sufficient condition for a beta-transformed geometric random variable to follow a power series distribution is derived. It is shown that a beta-transformed member of the Katz family does not belong to the Katz family unless it is a geometric distribution, thereby getting a characterization.

Dirichlet Random Walks

2014 ◽  
Vol 51 (04) ◽  
pp. 1081-1099 ◽  
Author(s):  
Gérard Letac ◽  
Mauro Piccioni
Keyword(s):  
Random Walk ◽  
Laplace Transform ◽  
Dirichlet Process ◽  
Hypergeometric Functions ◽  
Bessel Functions ◽  
Random Variable ◽  
Infinite Divisibility ◽  
Stieltjes Transform ◽  
The Laplace Transform ◽  
Infinite Divisible Distributions

This paper provides tools for the study of the Dirichlet random walk inRd. We compute explicitly, for a number of cases, the distribution of the random variableWusing a form of Stieltjes transform ofWinstead of the Laplace transform, replacing the Bessel functions with hypergeometric functions. This enables us to simplify some existing results, in particular, some of the proofs by Le Caër (2010), (2011). We extend our results to the study of the limits of the Dirichlet random walk when the number of added terms goes to ∞, interpreting the results in terms of an integral by a Dirichlet process. We introduce the ideas of Dirichlet semigroups and Dirichlet infinite divisibility and characterize these infinite divisible distributions in the sense of Dirichlet when they are concentrated on the unit sphere ofRd.

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