The joint distributions of running maximum of a Slepian processes

TL;DR

This paper derives explicit formulas for the joint distribution of the maximum values of a Slepian process over different intervals, providing insights into their probabilistic behavior and moments.

Contribution

It introduces explicit integral expressions for the joint distribution of partial and total maxima of a Slepian process, advancing understanding of their joint behavior.

Findings

Derived explicit integral formulas for the distribution of the partial maximum.

Obtained the joint distribution function of partial and total maxima.

Calculated moments of the partial maximum $m_s$.

Abstract

Consider the Slepian process $S$ defined by $ S(t)=B(t+1)-B(t),t\in [0,1]$ with $B(t),t\in \R$ a standard Brownian motion.In this contribution we analyze the joint distribution between the maximum $m_{s}=\max_{0\leq u\leq s}S(u)$ certain and the maximum $M_t=\max_{0\leq u\leq t}S(u)$ 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 $m_{s}$.