Refined properties for the Mellin transform, with applications to convergence of families obtained by biasing or by the stationary excess operator

TL;DR

This paper explores properties of the Mellin transform for nonnegative variables, formalizes convergence concepts, and applies these to analyze limit theorems involving size biasing and stationary excess operators, revealing necessary conditions and limit distributions.

Contribution

It introduces new properties of the Mellin transform, formalizes convergence in this context, and characterizes the limit distributions for sequences built via stationary excess operators and size biasing.

Findings

Continuous time convergence is equivalent to discrete time convergence.

Conditions by Harkness and Shantaram are necessary.

Limit distributions are mixtures of exponential and lognormal.

Abstract

We first provide some properties of the Mellin transform of nonnegative random variables, such that monotonicity, injectivity and effect of size biasing. Convergence of Mellin transforms is also entirely formalized through convergence in distribution and uniform integrability. As an application, we study a problem raised by Harkness and Shantaram (1969) who obtained, under sufficient conditions, a limit theorem for sequences of nonnegative random variables build with the iterated stationary excess operator. We reformulate this problem through the concept of multiply monotone functions and through the convergence of families build by the continuous time version of the iterated stationary excess operator and also by size biasing. The latter allows us to show that in our context, continuous time convergence is equivalent to discrete time convergence, that the conditions of Harkness and Shantaram are actually necessary and that the only possible limits in distribution are mixture of exponential with lognormal distributions.