Record statistics of a strongly correlated time series: random walks and Lévy flights

Abstract

Abstract. We review recent advances on the record statistics of strongly correlated time series, whose entries denote the positions of a random walk or a Lévy flight on a line. After a brief survey of the theory of records for independent and identically distributed random variables, we focus on random walks. During the last few years, it was indeed realized that random walks are a very useful "laboratory" to test the effects of correlations on the record statistics. We start with the simple one-dimensional random walk with symmetric jumps (both continuous and discrete) and discuss in detail the statistics of the number of records, as well as of the ages of the records, i.e., the lapses of time between two successive record breaking events. Then we review the results that were obtained for a wide variety of random walk models, including random walks with a linear drift, continuous time random walks, constrained random walks (like the random walk bridge) and the case of multiple independent random walkers. Finally, we discuss further observables related to records, like the record increments, as well as some questions raised by physical applications of record statistics, like the effects of measurement error and noise.arXiv:1702.00586v1 [cond-mat.stat-mech]

Statistics of occupation times and connection to local properties of nonhomogeneous random walks

“…For any symmetric jump distribution, one obtains the behavior Q n ∼ n −1/2 for large n, independently of the distribution itself. It can be shown that such a decay also holds for walks with nearest-neighbor jumps, i.e., a particular case of non-continuous jump distribution, provided that the jumps are symmetric, independent and identically distributed [32]. Therefore, in the paradigmatic case of the simple symmetric random walk on the integers, one finds that the value 1 2 describes the powerlaw decay of the survival probability and gives the correct parameter describing both the Lamperti and the Mittag-Leffler distributions.…”

Maximum-energy records in glassy energy landscapes

“…Using Eqs. (17), (18) and (25) together with the fact that inf I is negligible compared to E max (t), we get…”

“…In this regime Eqs. (17) and (18) still hold, whereas Eq. (7) fails, making E min (t) irrelevant for the calculation of E max (t).…”

Records for the moving average of a time series

“…. , I n are statistically independent [2,3,10]. The distribution of M n ensues from this fact by elementary considerations (see also section 7.4 below).…”