scholarly journals Return times for stochastic processes with power-law scaling

2007 ◽  
Vol 76 (1) ◽  
Author(s):  
Piero Olla
Keyword(s):  
Stochastic Processes ◽  
Power Law ◽  
Return Times

scholarly journals Potential Well in Poincaré Recurrence

Entropy ◽  
2021 ◽  
Vol 23 (3) ◽  
pp. 379
Author(s):  
Miguel Abadi ◽  
Vitor Amorim ◽  
Sandro Gallo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Stochastic Processes ◽  
Recurrence Time ◽  
Potential Well ◽  
Exponential Approximation ◽  
Mixing Processes ◽  
System Perspective ◽  
Return Times ◽  
Error Terms

From a physical/dynamical system perspective, the potential well represents the proportional mass of points that escape the neighbourhood of a given point. In the last 20 years, several works have shown the importance of this quantity to obtain precise approximations for several recurrence time distributions in mixing stochastic processes and dynamical systems. Besides providing a review of the different scaling factors used in the literature in recurrence times, the present work contributes two new results: (1) For ϕ-mixing and ψ-mixing processes, we give a new exponential approximation for hitting and return times using the potential well as the scaling parameter. The error terms are explicit and sharp. (2) We analyse the uniform positivity of the potential well. Our results apply to processes on countable alphabets and do not assume a complete grammar.

scholarly journals Effects of randomness on power law tails in multiplicatively interacting stochastic processes

2004 ◽  
Vol 324 (5-6) ◽  
pp. 378-382 ◽  
Author(s):  
Toshiya Ohtsuki ◽  
Akihiro Fujihara ◽  
Hiroshi Yamamoto
Keyword(s):  
Stochastic Processes ◽  
Power Law

Power-law autocorrelated stochastic processes with long-range cross-correlations

2007 ◽  
Vol 56 (1) ◽  
pp. 47-52 ◽  
Author(s):  
B. Podobnik ◽  
D. F. Fu ◽  
H. E. Stanley ◽  
P. Ch. Ivanov
Keyword(s):  
Stochastic Processes ◽  
Long Range ◽  
Power Law ◽  
Cross Correlations

Return times in nearly-completely decomposable stochastic processes

1978 ◽  
Vol 15 (02) ◽  
pp. 251-267 ◽  
Author(s):  
G. Latouche ◽  
G. Louchard
Keyword(s):  
Markov Chain ◽  
Stochastic Processes ◽  
Stochastic Matrix ◽  
Return Time ◽  
Second Order ◽  
Return Times ◽  
Second Moments

Consider a finite irreducible aperiodic Markov chain with nearly-completely decomposable stochastic matrix: i.e. a Markov chain for which the states can be grouped into disjoint aggregates, in such a way that the probabilities of transition between states of the same aggregate are large compared to the probabilities of transition between states belonging to different aggregates. Let Ω be a subset of one of the aggregates. Second-order approximations are determined for the first and second moments of the time to reach Ω and the return time to Ω.

scholarly journals Multifractal products of stochastic processes: construction and some basic properties

2002 ◽  
Vol 34 (4) ◽  
pp. 888-903 ◽  
Author(s):  
Petteri Mannersalo ◽  
Ilkka Norros ◽  
Rudolf H. Riedi
Keyword(s):  
Time Series ◽  
Stochastic Processes ◽  
Power Law ◽  
Special Form ◽  
Multifractal Spectrum ◽  
Nonstationary Process ◽  
Limit Process ◽  
Stationary Stochastic Processes ◽  
Measured Time ◽  
Multifractal Scaling

In various fields, such as teletraffic and economics, measured time series have been reported to adhere to multifractal scaling. Classical cascading measures possess multifractal scaling, but their increments form a nonstationary process. To overcome this problem, we introduce a construction of random multifractal measures based on iterative multiplication of stationary stochastic processes, a special form of T-martingales. We study the ℒ2-convergence, nondegeneracy, and continuity of the limit process. Establishing a power law for its moments, we obtain a formula for the multifractal spectrum and hint at how to prove the full formalism.

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