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We first provide some properties of the Mellin transform of nonnegativerandom variables, such that monotonicity, injectivity and effect of sizebiasing. Convergence of Mellin transforms is also entirely formalized throughconvergence in distribution and uniform integrability. As an application, westudy a problem raised by Harkness and Shantaram (1969) who obtained, undersufficient conditions, a limit theorem for sequences of nonnegative randomvariables build with the iterated stationary excess operator. We reformulatethis problem through the concept of multiply monotone functions and through theconvergence of families build by the continuous time version of the iteratedstationary excess operator and also by size biasing. The latter allows us toshow that in our context, continuous time convergence is equivalent to discretetime convergence, that the conditions of Harkness and Shantaram are actuallynecessary and that the only possible limits in distribution are mixture ofexponential with lognormal distributions.

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