We investigate extreme value statistics (EVS) of general discrete time and continuous space symmetric jump processes. We first show that for unbounded jump processes, the semi-infinite propagator $G_0(x,n)$, defined as the probability for a particle issued from zero to be at position $x$ after $n$ steps whilst staying positive, is the key ingredient needed to derive a variety of joint distributions of extremes and times at which they are reached. Along with exact expressions, we extract universal asymptotic behaviors of such quantities. For bounded, semi-infinite jump processes killed upon first crossing of zero, we introduce the strip probability ${\mu }_{0,\underline{x}}\left(n\right)$, defined as the probability that a particle issued from zero remains positive and reaches its maximum $x$ on its $n{\text{th}}^{}$ step exactly. We show that ${\mu }_{0,\underline{x}}\left(n\right)$ is the essential building block to address EVS of semi-infinite jump processes, and obtain exact expressions and universal asymptotic behaviors of various joint distributions.

Close

PrevNext

View figure in article

• Random walks
• Extreme event statistics
• Probability theory