Karhunen-Lo\`eve expansion for a generalization of Wiener bridge

TL;DR

This paper derives a Karhunen-Loève expansion for a generalized Gaussian process related to Wiener processes, which is significant in goodness-of-fit testing, including special cases like the Wiener bridge.

Contribution

It introduces a novel Karhunen-Loève expansion for a generalized Gaussian process involving a function g, extending previous results to broader classes of processes.

Findings

Derived explicit KL expansion for the process involving g

Special cases include the sine function and the Wiener bridge

Results applicable to goodness-of-fit test theory

Abstract

We derive a Karhunen-Lo\`eve expansion of the Gauss process $B_t - g(t)\int_0^1 g'(u)\,d B_u$, $t\in[0,1]$, where $(B_t)_{t\in[0,1]}$ is a standard Wiener process and $g:[0,1]\to R$ is a twice continuously differentiable function with $g(0) = 0$ and $\int_0^1 (g'(u))^2\,d u =1$. This process is an important limit process in the theory of goodness-of-fit tests. We formulate two special cases with the function $g(t)=\frac{\sqrt{2}}{\pi}\sin(\pi t)$, $t\in[0,1]$, and $g(t)=t$, $t\in[0,1]$, respectively. The latter one corresponds to the Wiener bridge over $[0,1]$ from $0$ to $0$.