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A note on moments of limit log-infinitely divisible stochastic measures of Bacry and Muzy

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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 (Commun 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.

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Notes

  1. What we call \(\mu \) is denoted \(\lambda ^2\) in [18]. Also, in [18] it is taken to be part of \(\phi (q),\) whereas we prefer to separate the two.

  2. The reader should note that other conical sets can be used to construct the measure. The other choices, however, lead to somewhat different properties of the limit measure, cf. [5] for a particular example.

  3. The nondegeneracy condition given in [4] is less stringent than Eq. (8), which is however sufficient in most cases of interest such as those of the limit lognormal, compound Poisson, etc. processes.

  4. By a slight abuse of terminology, we refer to any integral of the form \(\int _0^1 f(t)\,M_\mu (\mathrm{d}t)\) as the total mass.

  5. We first discovered these identities in the special case of the limit log-Poisson measure using the Almkvist–Zeilberger algorithm as implemented in the Maple package MultiAlmkvistZeilberger, cf. [1].

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Acknowledgements

The author wishes to thank Jakob Ablinger of the Research Institute for Symbolic Computation for verifying Eqs. (71)–(75) and attempting to compute the integrals in Eqs. (89) and (90) using an extension of the Almkvist-Zeilberger algorithm in his Mathematica package MultiIntegrate. The author gratefully acknowledges that he used Doron Zeilberger’s implementation of the same algorithm in the Maple package MultiAlmkvistZeilberger. We verified all of our results in Sects. 4 and 5 numerically using the computer algebra program MAXIMA (http://maxima.sourceforge.net). Finally, the author is thankful to the referee for several helpful suggestions.

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    Dmitry Ostrovsky

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Ostrovsky, D. A note on moments of limit log-infinitely divisible stochastic measures of Bacry and Muzy. Lett Math Phys 107, 267–289 (2017). https://doi.org/10.1007/s11005-016-0898-7

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