Record statistics and persistence for a random walk with a drift

Abstract

We study the statistics of records of a one-dimensional random walk of n steps, starting from the origin, and in presence of a constant bias c. At each time-step the walker makes a random jump of length ηdrawn from a continuous distribution f(η) which is symmetric around a constant drift c. We focus in particular on the case were f(η) is a symmetric stable law with a Lévy index 0 < μ\leq 2. The record statistics depends crucially on the persistence probability which, as we show here, exhibits different behaviors depending on the sign of c and the value of the parameter μ. Hence, in the limit of a large number of steps n, the record statistics is sensitive to these parameters (c and μ) of the jump distribution. We compute the asymptotic mean record number <R_n> after n steps as well as its full distribution P(R,n). We also compute the statistics of the ages of the longest and the shortest lasting record. Our exact computations show the existence of five distinct regions in the (c, 0 < μ\leq 2) strip where these quantities display qualitatively different behaviors. We also present numerical simulation results that verify our analytical predictions.

51 pages, 22 figures. Published version (typos have been corrected)

References in corpus (19)

• Persistence in Nonequilibrium Systems
• On the Role of Global Warming on the Statistics of Record-Breaking Temperatures
• Universal Record Statistics of Random Walks and Lévy Flights
• Canonical Analysis of Condensation in Factorised Steady State
• Driven particle in a random landscape: disorder correlator, avalanche distribution and extreme value statistics of records
• Record-breaking temperatures reveal a warming climate
• Record statistics for biased random walks, with an application to financial data
• Breaking records in the evolutionary race
• Record dynamics and the observed temperature plateau in the magnetic creep rate of type II superconductors
• Extreme value statistics from the Real Space Renormalization Group: Brownian Motion, Bessel Processes and Continuous Time Random Walks
• Records in a changing world
• Intermittent quakes and record dynamics in the thermoremanent magnetization of a spin-glass
• Records and sequences of records from random variables with a linear trend
• Record Statistics for Multiple Random Walks
• The longest excursion of stochastic processes in nonequilibrium systems
• A record-driven growth process
• Record Statistics of Continuous Time Random Walk
• Correlations of record events as a test for heavy-tailed distributions
• Correlations between record events in sequences of random variables with a linear trend

Cited by in corpus (43)

• Persistence and First-Passage Properties in Non-equilibrium Systems
• Records in stochastic processes -- Theory and applications
• Universal survival probability for a $d$-dimensional run-and-tumble particle
• Record statistics of a strongly correlated time series: random walks and Lévy flights
• Universal Properties of a Run-and-Tumble Particle in Arbitrary Dimension
• Record statistics for random walks and Lévy flights with resetting
• Extreme value statistics of ergodic Markov processes from first passage times in the large deviation limit
• Distribution of the Time Between Maximum and Minimum of Random Walks
• Survival Probability of Random Walks and Lévy Flights on a Semi-Infinite Line
• Splitting Probabilities of Jump Processes
• Universal statistics of longest lasting records of random walks and Lévy flights
• First-passage distributions for the one-dimensional Fokker-Planck equation
• Survival probability of a run-and-tumble particle in the presence of a drift
• Exact statistics of record increments of random walks and Lévy flights
• Statistics of the longest interval in renewal processes
• Universal framework for record ages under restart
• Optimization of random search processes in the presence of an external bias
• Statistics of the Number of Records for Random Walks and Lévy Flights on a $1D$ Lattice
• Record-breaking statistics for random walks in the presence of measurement error and noise
• Record breaking bursts during the compressive failure of porous materials
• Record statistics of financial time series and geometric random walks
• Modeling record-breaking stock prices
• Universal survival probability for a correlated random walk and applications to records
• Record statistics for random walk bridges
• Asymmetric Lévy flights in the presence of absorbing boundaries
• Smoluchowski flux and Lamb-Lion Problems for Random Walks and Lévy Flights with a Constant Drift
• Statistical properties of avalanches via the c-record process
• Controlling Uncertainty of Empirical First-Passage Times in the Small-Sample Regime
• Record Ages of Scale Invariant non-Markovian Random Walks
• Winding statistics of a Brownian particle on a ring
• Records for the number of distinct sites visited by a random walk on the fully-connected lattice
• Records in the classical and quantum standard map
• Time since maximum of Brownian motion and asymmetric Levy processes
• First-passage properties of the jump process with a drift. Two exactly solvable cases
• Exact and asymptotic properties of $\delta$-records in the linear drift model
• Extreme Event Statistics in a Drifting Markov Chain
• Record statistics for a discrete-time random walk with correlated steps
• Correspondence between noisy sample space reducing process and records in correlated random events
• Concentration of Empirical First-Passage Times
• Record breaking statistics near second order phase transitions
• First-passage properties of the jump process with a drift. The general case
• PhD thesis "Extreme value statistics of strongly correlated systems: fermions, random matrices and random walks"
• First-passage processes in a deterministic one-dimensional cellular automaton model of traffic flow