Karhunen-Loève expansions of mean-centered Wiener processes

Details Karhunen-Loève expansions of mean-centered Wiener processes Karhunen-Loève expansions of mean-centered Wiener processes

2006-12-22

arxiv.org/abs/math/0612693v1

IMS Lecture Notes Monograph Series 2006, Vol. 51, 62-76

Mathematics

Probability

Published at http://dx.doi.org/10.1214/074921706000000761 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/p

Scientific paper

10.1214/074921706000000761

For $\gamma>-{1/2}$, we provide the Karhunen-Lo\`{e}ve 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 $t^{\gamma}B(t)$.

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