On a Class of Distributions Stable Under Random Summation

We study a family of distributions that satisfy the stability-under-addition property, provided that the number ν of random variables in a sum is also a random variable. We call the corresponding property ν-stability and investigate the situation when the semigroup generated by the generating function of ν is commutative. Using results from the theory of iterations of analytic functions, we describe ν-stable distributions generated by summations with rational generating functions. A new case in this class of distributions arises when generating functions are linked with Chebyshev polynomials. The analogue of normal distribution corresponds to the hyperbolic secant distribution.

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Asymptotics of Bivariate Analytic Functions with Algebraic Singularities

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.6339 ◽

2020 ◽

Vol DMTCS Proceedings, 28th... ◽

Author(s):

Torin Greenwood

Keyword(s):

Generating Functions ◽

Analytic Functions ◽

Broad Class ◽

Multivariate Case ◽

Univariate Case ◽

International Audience ◽

Rational Generating Functions ◽

Algebraic Singularities

International audience In this paper, we use the multivariate analytic techniques of Pemantle and Wilson to find asymptotic for- mulae for the coefficients of a broad class of multivariate generating functions with algebraic singularities. Flajolet and Odlyzko (1990) analyzed the coefficients of a class of univariate generating functions with algebraic singularities. These results have been extended to classes of multivariate generating functions by Gao and Richmond (1992) and Hwang (1996, 1998), in both cases by immediately reducing the multivariate case to the univariate case. Pemantle and Wilson (2013) outlined new multivariate analytic techniques and used them to analyze the coefficients of rational generating functions.

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Bounds for moment generating functions and for extinction probabilities

Journal of Applied Probability ◽

10.2307/3212045 ◽

1966 ◽

Vol 3 (1) ◽

pp. 171-178 ◽

Cited By ~ 17

Author(s):

D. Brook

Keyword(s):

Generating Function ◽

Generating Functions ◽

Natural Extension ◽

Random Variable ◽

Moment Generating Function ◽

Multivariate Case ◽

Extinction Probabilities ◽

Moment Generating Functions

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

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Hopping from Chebyshev Polynomials to Permutation Statistics

The Electronic Journal of Combinatorics ◽

10.37236/8413 ◽

2019 ◽

Vol 26 (3) ◽

Author(s):

Jordan O. Tirrell ◽

Yan Zhuang

Keyword(s):

Generating Function ◽

Chebyshev Polynomials ◽

Generating Functions ◽

Permutation Statistics ◽

Domino Tilings ◽

Peak Number ◽

Exponential Generating Function

We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the (-1)-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic analogue of this action due to Sun and Wang.

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Bounds for moment generating functions and for extinction probabilities

Journal of Applied Probability ◽

10.1017/s0021900200114032 ◽

1966 ◽

Vol 3 (01) ◽

pp. 171-178 ◽

Cited By ~ 6

Author(s):

D. Brook

Keyword(s):

Generating Function ◽

Generating Functions ◽

Natural Extension ◽

Random Variable ◽

Moment Generating Function ◽

Multivariate Case ◽

Extinction Probabilities ◽

Moment Generating Functions

Suppose that we have a non-negative, real valued random variable x, whose distribution is governed by some unknown moment generating function M(t). Suppose further that we are given certain moments of x, then the question to be discussed in this paper is : can we find a sharp upper bounding function for the m.g.f.? It will be shown that this is usually possible both in the single variate case and in its natural extension to the multivariate case.

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A General Family of q-Hypergeometric Polynomials and Associated Generating Functions

Mathematics ◽

10.3390/math9111161 ◽

2021 ◽

Vol 9 (11) ◽

pp. 1161

Author(s):

Hari Mohan Srivastava ◽

Sama Arjika

Keyword(s):

Generating Function ◽

Generating Functions ◽

Hypergeometric Functions ◽

Additional Parameter ◽

Physical Sciences ◽

General Family ◽

Hypergeometric Polynomials

Basic (or q-) series and basic (or q-) polynomials, especially the basic (or q-) hypergeometric functions and the basic (or q-) hypergeometric polynomials are studied extensively and widely due mainly to their potential for applications in many areas of mathematical and physical sciences. Here, in this paper, we introduce a general family of q-hypergeometric polynomials and investigate several q-series identities such as an extended generating function and a Srivastava-Agarwal type bilinear generating function for this family of q-hypergeometric polynomials. We give a transformational identity involving generating functions for the generalized q-hypergeometric polynomials which we have introduced here. We also point out relevant connections of the various q-results, which we investigate here, with those in several related earlier works on this subject. We conclude this paper by remarking that it will be a rather trivial and inconsequential exercise to give the so-called (p,q)-variations of the q-results, which we have investigated here, because the additional parameter p is obviously redundant.

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Binomial subsampling

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2005.1622 ◽

2006 ◽

Vol 462 (2068) ◽

pp. 1181-1195 ◽

Cited By ~ 12

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.

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