Record statistics of continuous time random walk

Raman Research Institute - Bangalore 560080, India

^a sanjib@rri.res.in

Received: 28 December 2010
Accepted: 10 March 2011

Abstract

The statistics of records for a time series generated by a continuous time random walk is studied, and found to be independent of the details of the jump length distribution, as long as the latter is continuous and symmetric. However, the statistics depend crucially on the nature of the waiting-time distribution. The probability of finding M records within a given time duration t, for large t, has a scaling form, and the exact scaling function is obtained in terms of the one-sided Lévy stable distribution. The mean of the ages of the records, defined as ⟨t/M⟩, differs from t/⟨M⟩. The asymptotic behaviour of the shortest and the longest ages of the records are also studied.