Distribution of the Time Between Maximum and Minimum of Random Walks

Abstract

We consider a one-dimensional Brownian motion of fixed duration $T$. Using a path-integral technique, we compute exactly the probability distribution of the difference $\tau =t_{\min }-t_{\max }$ between the time $t_{\min }$ of the global minimum and the time $t_{\max }$ of the global maximum. We extend this result to a Brownian bridge, i.e. a periodic Brownian motion of period $T$. In both cases, we compute analytically the first few moments of $\tau$, as well as the covariance of $t_{\max }$ and $t_{\min }$, showing that these times are anti-correlated. We demonstrate that the distribution of $\tau$ for Brownian motion is valid for discrete-time random walks with $n$ steps and with a finite jump variance, in the limit $n\to \infty$. In the case of Lévy flights, which have a divergent jump variance, we numerically verify that the distribution of $\tau$ differs from the Brownian case. For random walks with continuous and symmetric jumps we numerically verify that the probability of the event "$\tau =n$" is exactly $1/(2n)$ for any finite $n$, independently of the jump distribution. Our results can be also applied to describe the distance between the maximal and minimal height of $(1+1)$-dimensional stationary-state Kardar-Parisi-Zhang interfaces growing over a substrate of finite size $L$. Our findings are confirmed by numerical simulations. Some of these results have been announced in a recent Letter [Phys. Rev. Lett. 123, 200201 (2019)].

arXiv comment33 pages, 12 figures. This is a longer version of arXiv:1909.05594, published in Physical Review Letters. Typos corrected

References in corpus (33)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Extreme value statistics of correlated random variables: a pedagogical review
• Airy Distribution Function: From the Area Under a Brownian Excursion to the Maximal Height of Fluctuating Interfaces
• Exact Maximal Height Distribution of Fluctuating Interfaces
• Universal First-passage Properties of Discrete-time Random Walks and Levy Flights on a Line: Statistics of the Global Maximum and Records
• On the Role of Global Warming on the Statistics of Record-Breaking Temperatures
• Universal Record Statistics of Random Walks and Lévy Flights
• Random Convex Hulls and Extreme Value Statistics
• Record-breaking temperatures reveal a warming climate
• Exact solutions for the statistics of extrema of some random 1D landscapes, Application to the equilibrium and the dynamics of the toy model
• On the time to reach maximum for a variety of constrained Brownian motions
• Safe Leads and Lead Changes in Competitive Team Sports
• Record statistics and persistence for a random walk with a drift
• Generalized arcsine laws for fractional Brownian motion
• Universal Asymptotic Statistics of Maximal Relative Height in One-dimensional Solid-on-solid Models
• Hitting probability for anomalous diffusion processes
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Generalised `Arcsine' laws for run-and-tumble particle in one dimension
• Traveling Waves, Front Selection, and Exact Nontrivial Exponents in a Random Fragmentation Problem
• Records in a changing world
• Time to reach the maximum for a random acceleration process
• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Distribution of the time at which N vicious walkers reach their maximal height
• Extreme-Value Statistics of Fractional Brownian Motion Bridges
• Record occurrence and record values in daily and monthly temperatures
• Exact distributions of the number of distinct and common sites visited by N independent random walkers
• Time between the maximum and the minimum of a stochastic process
• Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time
• An exactly solvable record model for rainfall
• On the Gap and Time Interval between the First Two Maxima of Long Random Walks
• Exact statistics of record increments of random walks and Lévy flights
• Arcsine Laws in Stochastic Thermodynamics
• Exact distributions of cover times for $N$ independent random walkers in one dimension

Cited by in corpus (16)

• Extremal statistics for stochastic resetting systems
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Mean perimeter and area of the convex hull of a planar Brownian motion in the presence of resetting
• Learning physical properties of anomalous random walks using graph neural networks
• Distribution of the time of the maximum for stationary processes
• Number of distinct sites visited by a resetting random walker
• Extreme value statistics and arcsine laws for heterogeneous diffusion processes
• Time to reach the maximum for a stationary stochastic process
• Extremal statistics of a one dimensional run and tumble particle with an absorbing wall
• Extreme Value Statistics and Arcsine Laws of Brownian Motion in the Presence of a Permeable Barrier
• Extremal statistics for a resetting Brownian motion before its first-passage time
• Extremal statistics for a one-dimensional Brownian motion with a reflective boundary
• Generalized arcsine laws for a sluggish random walker with subdiffusive growth
• Randomness of Mobius coefficents and brownian motion: growth of the Mertens function and the Riemann Hypothesis
• Exact joint distributions of three global characteristic times for Brownian motion
• Sampling rare trajectories using stochastic bridges