Birth Processes and Symmetric Polynomials

Abstract

Many discrete-time pure birth processes are symmetrical in the following sense: The probability that the process is in a fixed state is independent of the sequence of transitions inducing . This is always the case whenever a transition is a time-dependent or a state-dependent random variable or the product of such independent variables. We use this property in order to derive algebraic descriptions in terms of symmetric polynomials. Besides new solutions, our approach offers a uniform point of view on a large class of often considered distributions.

The Asymptotic Behavior of Certain Birth Processes

“…(2.1) Since our notation is slightly different from [2], we provide a translation for the reader. Our transition probability τ m, n in formula (1.1) corresponds λ n, m in [2, 1.3].…”

“…Our work on this topic was motivated by the results of [2]. Our Theorem 1.3 is a far-reaching generalization of Theorems 3.2 -3.4 of that paper.…”

The Group of Generalized Stirling Numbers

“…Finally there are authors like D'Antona, Damiani, Hsu, Loeb, Naldi, and Shiue [5][6][7] who have looked at GSN basically as connection constants arising from the transformation between sequences of polynomials with persistent roots. Recently Bickel et al [2] have shown that the GSN are well suited for the description of discrete time pure birth processes.…”

Some characteristic properties of the weighted particular Schur polynomial mean

“…For the basic properties of the symmetric polynomials, see Macdonald. 5 Some recent studies on symmetric polynomials include, for example, literature [17][18][19][20][21][22][23][24][25] (see also previous works [26][27][28][29][30][31][32] ).…”