Karhunen-Lo\`{e}ve expansions of mean-centered Wiener processes

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 $t^{\gamma}B(t)$.