The Cameron-Martin Theorem for (p-)Slepian processes

TL;DR

This paper establishes a Cameron-Martin theorem for p-Slepian processes, providing explicit density formulas and identifying the class of functions for which these densities exist, with applications in scan statistics.

Contribution

It extends the Cameron-Martin theorem to p-Slepian processes and derives explicit density formulas, a novel result in the context of these processes.

Findings

Derived the class of functions with existing densities for p-Slepian processes

Provided explicit formulas for the densities of shifted p-Slepian processes

Linked p-Slepian processes to applications in scan statistics and parameter testing

Abstract

We show a Cameron-Martin theorem for Slepian processes $W_t:=\frac{1}{\sqrt{p}}(B_t-B_{t-p}), t\in [p,1]$, where $p\geq \frac{1}{2}$ and $B_s$ is Brownian motion. More exactly, we determine the class of functions $F$ for which a density of $F(t)+W_t$ with respect to $W_t$ exists. Moreover, we prove an explicit formula for this density. p-Slepian processes are closely related to Slepian processes. p-Slepian processes play a prominent role among others in scan statistics and in testing for parameter constancy when data are taken from a moving window.