• Bernoulli
  • Volume 22, Number 4 (2016), 2579-2608.

Limit theorems for multifractal products of geometric stationary processes

Denis Denisov and Nikolai Leonenko

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We investigate the properties of multifractal products of geometric Gaussian processes with possible long-range dependence and geometric Ornstein–Uhlenbeck processes driven by Lévy motion and their finite and infinite superpositions. We present the general conditions for the $\mathcal{L}_{q}$ convergence of cumulative processes to the limiting processes and investigate their $q$th order moments and Rényi functions, which are non-linear, hence displaying the multifractality of the processes as constructed. We also establish the corresponding scenarios for the limiting processes, such as log-normal, log-gamma, log-tempered stable or log-normal tempered stable scenarios.

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Bernoulli, Volume 22, Number 4 (2016), 2579-2608.

Received: October 2013
Revised: May 2015
First available in Project Euclid: 3 May 2016

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geometric Gaussian process geometric Ornstein–Uhlenbeck processes Lévy processes log-gamma scenario log-normal scenario log-normal tempered stable scenario long-range dependence log-variance gamma scenario multifractal products multifractal scenarios Rényi function scaling of moments short-range dependence stationary processes superpositions


Denisov, Denis; Leonenko, Nikolai. Limit theorems for multifractal products of geometric stationary processes. Bernoulli 22 (2016), no. 4, 2579--2608. doi:10.3150/15-BEJ738.

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