On the Limit Lognormal and Other Limit Log-Infinitely Divisible Laws

Abstract

The limit log-infinitely divisible multifractals of Muzy and Bacry (Phys. Rev. E 66:056121, 2002) are reviewed and shown to possess novel invariance relations that translate into functional Feynman-Kac equations for the corresponding probability distributions. In the special case of the limit lognormal process of Mandelbrot (in Statistical Models and Turbulence, M. Rosenblatt, C. Van Atta (Eds.), Springer, New York, 1972), the limit distribution is represented exactly in an operator form using the technique of intermittency expansions. A novel representation for the Mellin transform of the limit distribution is derived and related to the Hurwitz zeta function. For application, the cumulants of the logarithm of the limit lognormal distribution are computed explicitly.