Record Statistics of Integrated Random Walks and the Random Acceleration Process
Abstract
We address the theory of records for integrated random walks with finite variance. The longtime continuum limit of these walks is a nonMarkov process known as the random acceleration process or the integral of Brownian motion. In this limit, the renewal structure of the record process is the cornerstone for the analysis of its statistics. We thus obtain the analytical expressions of several characteristics of the process, notably the distribution of the total duration of record runs (sequences of consecutive records), which is the continuum analogue of the number of records of the integrated random walks. This result is universal, i.e., independent of the details of the parent distribution of the step lengths.
 Publication:

Journal of Statistical Physics
 Pub Date:
 January 2022
 DOI:
 10.1007/s10955021028529
 arXiv:
 arXiv:2109.05582
 Bibcode:
 2022JSP...186....4G
 Keywords:

 Records;
 Integrated random walks;
 Integrated Brownian motion;
 Random acceleration process;
 Renewal theory;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Probability
 EPrint:
 32 pages, 7 figures