Karhunen-Loève expansions of mean-centered Wiener processes

Abstract

For $\gamma >-1/2$, we provide the Karhunen-Loève expansion of the weighted mean-centered Wiener process, defined by \[W _γ(t)=\frac{1}{\sqrt{1+2γ}}\Big\{W\big(t^{1+2γ}\big)- \int_0^1W\big(u^{1+2γ}\big)du\Big\},\] 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 $t^{\gamma }B(t)$.

Published at http://dx.doi.org/10.1214/074921706000000761 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org)