Probability Generating Functions for Discrete Real Valued Random Variables

The probability generating function is a powerful technique for studying the law of finite sums of independent discrete random variables taking integer positive values. For real valued discrete random variables, the well known elementary theory of Dirichlet series and the symbolic computation packages available nowadays, such as Mathematica 5 TM, allows us to extend to general discrete random variables this technique. Being so, the purpose of this work is twofold. Firstly we show that discrete random variables taking real values, non necessarily integer or rational, may be studied with probability generating functions. Secondly we intend to draw attention to some practical ways of performing the necessary calculations. 1. Classical probability generating functions Generating functions are an useful and up to date tool in nowadays practical mathematics, in particular in discrete mathematics and combinatorics (see [Lando 03]) and, in the case of probability generating functions, in distributional convergence results as in [Kallenberg 02][p. 84]. Its uses in basic probability are demonstrated in the classic reference [Feller 68][p. 266]. More recently, probability generating functions for integer valued random variables have been studied intensively mainly with some applied purposes in mind. See, for instance [Dowling et al 97], [Marques et al 89], [Nakamura et al 93], [Nakamura et al 93a], [Nakamura et al 93b], [Rémillard et al 00], [Rueda et al 91] and [Rueda et al 99]. The natural extension of the definition of probability generating function to non negative real valued random variable X, as the expectation of the function tX , is very clearly presented in the treatise [Hoffmann-Jørgensen 94][p. 288] where some of the consequences, drawn directly from this definition, are stated. Let us briefly formulate some classical results on probability generating functions for integer valued random variables recalling the useful connection between the topics of probability generating functions and of analytic function theory. Let X be a random variable taking values in Z and consider that for all k ∈ Z we have pk := P[X = k] ∈ [0, 1]. Date: July 19, 2004. 1991 Mathematics Subject Classification. 60–08, 60E10, 30B50.