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

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 [$\text{Comm. Math. Phys.}$ $236$: 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.

19 pages. To appear in Lett. Math. Phys

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• Freezing and extreme value statistics in a Random Energy Model with logarithmically correlated potential
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Cited by in corpus (2)

• A Review of Conjectured Laws of Total Mass of Bacry-Muzy GMC Measures on the Interval and Circle and Their Applications
• A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures