Distribution of the time of the maximum for stationary processes

We consider a one-dimensional stationary stochastic process x(tau) of duration T. We study the probability density function (PDF) P(t(m)vertical bar T) of the time t(m) at which x(tau) reaches its global maximum. By using a path integral method, we compute P(t(m)vertical bar 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(m)vertical bar 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 << t(m) << T and has a nontrivial edge scaling behavior for t(m) -> 0 (and when t(m) -> T), that we compute exactly. Moreover, we show that for any equilibrium process the PDF P(t(m)vertical bar T) is symmetric around t(m) = T/2, i.e., P(t(m)vertical bar T) = P(T - t(m)vertical bar T). This symmetry provides a simple method to decide whether a given stationary time series x(tau) is at equilibrium or not. Copyright (C) 2021 EPLA