Mellin Transform of the Limit Lognormal Distribution

摘要

The technique of intermittency expansions is applied to derive an exact formal power series representation for the Mellin transform of the probability distribution of the limit lognormal multifractal process. The negative integral moments are computed by a novel product formula of Selberg type. The power series is summed in general by means of its small intermittency asymptotic. The resulting integral formula for the Mellin transform is conjectured to be valid at all levels of intermittency. The conjecture is verified partially by proving that the integral formula reproduces known results for the positive and negative integral moments of the limit lognormal distribution and gives a valid characteristic function of the Lévy-Khinchine type for the logarithm of the distribution. The moment problem for the logarithm of the distribution is shown to be determinate, whereas the moment problems for the distribution and its reciprocal are shown to be indeterminate. The conjecture is used to represent the Mellin transform as an infinite product of gamma factors generalizing Selberg’s finite product. The conjectured probability density functions of the limit lognormal distribution and its logarithm are computed numerically by the inverse Fourier transform. Communicated by S. Zelditch