Article

Correlation cascades of Lévy-driven random processes

School of Chemistry and School of Mathematics, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel; School of Chemistry, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel
Physica A: Statistical Mechanics and its Applications DOI:10.1016/j.physa.2006.10.029 pp.1-26

ABSTRACT We explore the correlation-structure of a large class of random processes, driven by non-Gaussian Lévy noise sources with possibly infinite variances. Examples of such processes include Lévy motions, Lévy-driven Ornstein–Uhlenbeck motions, Lévy-driven moving-average processes, fractional Lévy motions, and fractional Lévy noises.Based on the fact that non-Gaussian Lévy noises are continuum superpositions of Poisson noises, we unveil an underlying Cascade of ‘Lévy correlation functions’ which characterize the process-distribution and the correlation-structure of the processes under consideration. In the case where the driving Lévy noise sources are ‘fractal’, the resulting cascade admits a unique scale-free form.

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    Article: Correlation cascades, ergodic properties and long memory of infinitely divisible processes
    Marcin Magdziarz
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    ABSTRACT: In this paper, we investigate the properties of the recently introduced measure of dependence called correlation cascade. We show that the correlation cascade is a promising tool for studying the dependence structure of infinitely divisible processes. We describe the ergodic properties (ergodicity, weak mixing, mixing) of stationary infinitely divisible processes in the language of the correlation cascade and establish its relationship with the codifference. Using the correlation cascade, we investigate the dependence structure of four fractional [alpha]-stable stationary processes. We detect the property of long memory and verify the ergodic properties of the discussed processes.
    Stochastic Processes and their Applications 119(10):3416-3434. · 1.01 Impact Factor
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Keywords

correlation-structure
 
driving Lévy noise sources
 
fractional Lévy motions
 
fractional Lévy noises.Based
 
infinite variances
 
large class
 
Lévy motions
 
Lévy-driven moving-average processes
 
Lévy-driven Ornstein–Uhlenbeck motions
 
non-Gaussian Lévy noise sources
 
non-Gaussian Lévy noises
 
Poisson noises
 
processes
 
random processes
 
resulting cascade
 
underlying Cascade
 
unique scale-free form
 
‘Lévy correlation functions’