Ages of Records in Random Walks

Extreme value functionals of stochastic processes are inverse functionals of the first passage time -a connection that renders their probability distribution functions equivalent. Here, we deepen this link and establish a framework for analyzing extreme value statistics of ergodic reversible Markov processes in confining potentials on the hand of the underlying relaxation eigenspectra. We derive a chain of inequalities, which bounds the long-time asymptotics of first passage densities, and thereby extrema, from above and from below. The bounds involve a time integral of the transition probability density describing the relaxation towards equilibrium. We apply our general results to the analysis of extreme value statistics at long times in the case of Ornstein-Uhlenbeck process and a 3-dimensional Brownian motion confined to a sphere, also known as Bessel process. We find that even on time-scales that are shorter than the equilibration time, the large deviation limit characterizing long-time asymptotics can approximate the statistics of extreme values remarkably well. Our findings provide a novel perspective on the study of extrema beyond the established limit theorems for sequences of independent random variables and for asymmetric diffusion processes beyond a constant drift.

Section: Statistics Of Maxima In the Ornstein Uhlenbeck Processmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit

Hartich

Godec

2019

“…. , (M ) N /N ) converges, in the large N limit, to PD(1/2, 0) [117]. There is no simple explicit expression for the Poisson-Dirichlet distributions PD(α, θ) but ref [58] provided various probabilistic interpretations and constructions of this joint law.…”

Section: Joint Distribution Of the Ranked Ages: Poisson-dirichlet Dismentioning

confidence: 99%

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

Godrèche

Majumdar

Schehr

2017

132

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]

“…One can also note that the averages R (k)III sum up to t N /t and that t N /t → θ [2]. (See also [14,5]. )…”

Section: Average Longest Intervalmentioning

confidence: 99%

Longest interval between zeros of the tied-down random walk, the Brownian bridge and related renewal processes

Godrèche¹

2017

The probability distribution of the longest interval between two zeros of a simple random walk starting and ending at the origin, and of its continuum limit, the Brownian bridge, was analysed in the past by Rosén and Wendel, then extended by the latter to stable processes. We recover and extend these results using simple concepts of renewal theory, which allows to revisit past or recent works of the physics literature.