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

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.