Extreme value statistics of jump processes

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 μ_{0,[under x]̲}(n), defined as the probability that a particle issued from zero remains positive and reaches its maximum x on its nth^{} step exactly. We show that μ_{0,[under x]̲}(n) 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.