Karhunen-Loève expansion for a generalization of Wiener bridge

Abstract

We derive a Karhunen-Loève expansion of the Gauss process $B_t-g(t){\int }_0^1{g}^{\prime }(u)d{B}_u$, $t\in [0,1]$, 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 ${\int }_0^1(g^{\prime }(u){)}^{2}du=1$. 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 $0$.

arXiv comment25 pages, 1 figure. The appendix is extended