On the random gamma function: Theory and computing

Abstract

This paper deals with the extension, in the mean square sense, of the deterministic gamma function to the random framework. In a first step, we provide such extension to Γ(A) by assuming that the parameter A is a positive random variable satisfying certain conditions related to its exponential moments. As a particular case, we show that every positive random variable satisfies such conditions if it is bounded and bounded away from zero. In a second step, we establish the formula Γ(A+1)=AΓ(A) that allows us to extend the random gamma function to a class of random variables whose supports lie over the real line with the exception of small neighborhoods of zero and of the negative integers. This retains the classical definition of the gamma function when A becomes a deterministic parameter. The study is based on the Lp stochastic calculus with p=2 and 4, usually referred to as mean square and mean fourth stochastic calculus, respectively. Next, we compute the mean and the variance of the random gamma function, including several illustrative examples. Finally, with the aid of the random gamma function, we define the random Bessel function and compute reliable approximations of its mean and variance.