Distribution of the time of the maximum for stationary processes

Abstract

We consider a one-dimensional stationary stochastic process of duration T. We study the probability density function (PDF) of the time at which reaches its global maximum. By using a path integral method, we compute 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 , 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 and has a nontrivial edge scaling behavior for (and when ), that we compute exactly. Moreover, we show that for any equilibrium process the PDF is symmetric around , i.e., . This symmetry provides a simple method to decide whether a given stationary time series is at equilibrium or not.

Time Between the Maximum and the Minimum of a Stochastic Process

“…Mat. (SM) [38]. One then computes the probability of the trajectory for a fixed and eventually takes the limit → 0.…”

“…Using the Green's function in Eq. ( 9), one can compute explicitly this grand PDF and finally integrate over x max and x min to obtain the joint PDF P (t min , t max |T ) [see SM [38] for details]. From the latter, we compute the covariance of t min and t max explicitly, as given in Eq.…”

“…Also, by integrating the joint PDF P (t min , t max |T ) over t max and t min with t min − t max = τ fixed, we compute explicitly the PDF P (τ |T ) and find that it has the scaling form given in Eq. ( 4) [38].…”

“…To investigate this question, we computed P (τ |n) exactly for the special of a double exponential jump PDF p(η) = (1/2)e −|η| . The details are left in the SM [38] and here we just outline the main results. We find that, for large n, P (τ, |n) converges to the scaling form P (τ, |n) ≈ (1/n)f exp (τ /n) where f exp (y) satisfies the integral equation (10) Remarkably, this integral equation can be solved and we show that f exp (y) = f BM (y) obtained for the continuoustime BM in Eq.…”

“…This representation of f BM (y) in Eq. ( 10) turns out to be useful to compute the moments of τ explicitly (see [38]). We would indeed expect this Brownian scaling form to hold for any jump densities with a finite variance σ 2 , due to the Central Limit Theorem.…”