Distribution of the time of the maximum for stationary processes

Abstract

We consider a one-dimensional stationary stochastic process $x(\tau )$ of duration $T$. We study the probability density function (PDF) $P(t_{\mathrm{m}}|T)$ of the time $t_{\mathrm{m}}$ at which $x(\tau )$ reaches its global maximum. By using a path integral method, we compute $P(t_{\mathrm{m}}|T)$ for a number of equilibrium and nonequilibrium stationary processes, including the Ornstein-Uhlenbeck process, Brownian motion with stochastic resetting and a single confined run-and-tumble particle. For a large class of equilibrium stationary processes that correspond to diffusion in a confining potential, we show that the scaled distribution $P(t_{\mathrm{m}}|T)$, for large $T$, has a universal form (independent of the details of the potential). This universal distribution is uniform in the ``bulk'', i.e., for $0\ll t_{\mathrm{m}}\ll T$ and has a nontrivial edge scaling behavior for $t_{\mathrm{m}}\to 0$ (and when $t_{\mathrm{m}}\to T$), that we compute exactly. Moreover, we show that for any equilibrium process the PDF $P(t_{\mathrm{m}}|T)$ is symmetric around $t_{\mathrm{m}}=T/2$, i.e., $P(t_{\mathrm{m}}|T)=P(T-t_{\mathrm{m}}|T)$. This symmetry provides a simple method to decide whether a given stationary time series $x(\tau )$ is at equilibrium or not.

arXiv commentMain text: 6 pages + 4 figs., Supp. Mat.: 7 pages + 3 figs

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