A Note on Moments of Limit Log Infinitely Divisible Stochastic Measures of Bacry and Muzy

TL;DR

This paper derives a multiple integral representation for moments of the limit log-infinitely divisible measure, explores its covariance structure, and connects moments to generalized Selberg integrals, with applications to specific measures.

Contribution

It introduces a new integral representation for moments of the measure and generalizes the Selberg integral to analyze joint moments and covariance structure.

Findings

Covariance of total mass is logarithmic.

Derived multiple integral representation for moments.

Connected moments to generalized Selberg and Morris integrals.

Abstract

A multiple integral representation of single and joint moments of the total mass of the limit log-infinitely divisible stochastic measure of Bacry and Muzy [$\textit{Comm. Math. Phys.}$ ${\bf 236}$: 449-475, 2003] is derived. The covariance structure of the total mass of the measure is shown to be logarithmic. A generalization of the Selberg integral corresponding to single moments of the limit measure is proposed and shown to satisfy a recurrence relation. The joint moments of the limit lognormal measure, classical Selberg integral with $\lambda_1=\lambda_2=0,$ and Morris integral are represented in the form of multiple binomial sums. For application, low moments of the limit log-Poisson measure are computed exactly and low joint moments of the limit lognormal measure are considered in detail.