Correlated continuous time random walks

Mark M. Meerschaert, Erkan Nane and Yimin Xiao

Statistics & Probability Letters, 2009, vol. 79, issue 9, 1194-1202

Abstract: Continuous time random walks impose a random waiting time before each particle jump. Scaling limits of heavy-tailed continuous time random walks are governed by fractional evolution equations. Space-fractional derivatives describe heavy-tailed jumps, and the time-fractional version codes heavy-tailed waiting times. This paper develops scaling limits and governing equations in the case of correlated jumps. For long-range dependent jumps, this leads to fractional Brownian motion or linear fractional stable motion, with the time parameter replaced by an inverse stable subordinator in the case of heavy-tailed waiting times. These scaling limits provide an interesting class of non-Markovian, non-Gaussian self-similar processes.

Date: 2009
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