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VOL. 51 | 2006 Karhunen-Loève expansions of mean-centered Wiener processes

## Abstract

For $\gamma>-\frac{1}{2}$, we provide the Karhunen-Loève expansion of the weighted mean-centered Wiener process, defined by $W_{\gamma}(t)=\frac{1}{\sqrt{1+2\gamma}}\Big\{W\big(t^{1+2\gamma}\big) -\int_{0}^1W\big(u^{1+2\gamma}\big)du\Big\},$ for $t\in(0,1]$. We show that the orthogonal functions in these expansions have simple expressions in term of Bessel functions. Moreover, we obtain that the $L^2[0,1]$ norm of $W_{\gamma}$ is identical in distribution with the $L^2[0,1]$ norm of the weighted Brownian bridge $t^{\gamma}B(t)$.

## Information

Published: 1 January 2006
First available in Project Euclid: 28 November 2007

zbMATH: 1130.60045
MathSciNet: MR2387761

Digital Object Identifier: 10.1214/074921706000000761

Subjects:
Primary: 62G10
Secondary: 60F15 , 60G15 , 60H07 , 62G30

Keywords: Brownian bridge , Gaussian processes , Karhunen-Loeve expansions , quadratic functionals , Wiener process