Log-infinitely divisible multifractal processes

We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal \"Multifractal Random Walk\" processes (MRW) and the log...

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Bibliographic Details
Published in:Communications in mathematical physics 2003-06, Vol.236 (3), p.449-475
Main Authors: BACRY, E, MUZY, J. F
Format: Article
Language:English
Subjects:
Condensed Matter
Exact sciences and technology
Inference from stochastic processes
Inference from stochastic processes
time series analysis
Mathematical methods in physics
Mathematics
Physics
Probability and statistics
Probability theory and stochastic processes
Probability theory, stochastic processes, and statistics
Sciences and techniques of general use
Statistical Mechanics
Statistics
Stochastic processes
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Summary:We define a large class of multifractal random measures and processes with arbitrary log-infinitely divisible exact or asymptotic scaling law. These processes generalize within a unified framework both the recently defined log-normal \"Multifractal Random Walk\" processes (MRW) and the log-Poisson \"product of cynlindrical pulses\". Their construction involves some ``continuous stochastic multiplication\'\' from coarse to fine scales. They are obtained as limit processes when the finest scale goes to zero. We prove the existence of these limits and we study their main statistical properties including non degeneracy, convergence of the moments and multifractal scaling.
ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-003-0827-3