Limit theorems for continuous time random walks with slowly varying waiting times
type: Item
instance of: scholarly article
title: Limit theorems for continuous time random walks with slowly varying waiting times
zbMATH Open document ID: 1065.60030
author: Mark M. Meerschaert
author: Hans-Peter Scheffler
published in: Statistics \& Probability Letters
publication date: 2005-03-08T00:00:00Z
review text: Continuous time random walks are used in physics to model particle motion assuming random waiting times between the random jumps of a particle. The authors introduce the new feature of i.i.d. waiting times with a slowly varying probability tail. In this case the asymptotic waiting time process is known to be a Fréchet-type extremal process. In a corresponding limit theorem, the hitting time process is shown to have exponentially distributed marginals and increments with a defective exponential distribution. For an operator stable limit process of the i.i.d. jumps, assuming jumps and waiting times to be independent, an application of an appropriate transfer theorem proves a functional limit theorem for the continuous time random walk. Unlike the case of regular varying power law tails of the waiting times, the defectiveness of hitting time increments implies that particles may rest for a random time period in the resulting limiting process. The presented results are thus useful in physics to model ultraslow diffusions.
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60G52
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 2142302
zbMATH Keywords: anomalous diffusion
zbMATH Keywords: continuous time random walk
zbMATH Keywords: operator stable law
zbMATH Keywords: fractional derivative
MaRDI profile type: Publication
full work available at URL: j.spl.2004.10.030
OpenAlex ID: W2009984798
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DOI: 10.1016/J.SPL.2004.10.030
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description: scientific article; zbMATH DE number 2142302