The Joint Distribution of Running Maximum of a Slepian Process

Abstract

Consider the Slepian process S defined by S(t) = B(t + 1) − B(t), t ∈ [0, 1] with B(t), t ∈ R a standard Brownian motion. In this contribution we analyze the joint distribution between the maximum m s = max 0≤u≤s S(u) certain and the maximum M t = max 0≤u≤t S(u) for 0 < s < t fixed. Explicit integral expression are obtained for the distribution function of the partial maximum m s and the joint distribution function between m s and M t . We also use our results to determine the moments of m s .

Covariance of the running range of a Brownian trajectory

“…In contrast, the question how the extremes of BM are correlated in time has received essentially less attention. Only within the last few years, the two-time correlations of the running maximum of BM [30], of a Brownian Bridge (a BM constrained to return to the starting point at a fixed time moment t) [31] , the temporal correlations of a Slepian process (the difference of two BM positions taken at different time moments) [32] and of some records related to BM [33,34] have been determined. Here we focus on the two-time correlations (the covariance function) of the running range of a Brownian motion (see Fig.…”

“…On the other hand, the question how the extreme values of BM are correlated in time has received essentially less attention, although for many basic stochastic processes this issue has been usually addressed at the first place. This problem was approached first only within the past few years, and the two-time correlations of the running maximum of BM [35], of a Brownian bridge (a BM constrained to return to the starting point at a fixed time moment t) [36], and of a Slepian process (the difference of two BM positions taken at different time moments) [37], as well as of some records related to BM [11,38,39] have been determined. Here we focus on the two-time correlations (the covariance function) of the running range of Brownian motion (see figure 1)-a maximal extent (span) of a BM on a given time interval.…”