A Karhunen-Loeve expansion for a mean-centered Brownian bridge

The processes of the form , where K is a constant, and B(·) a Brownian bridge, are investigated. We show that and are both Brownian bridges, and establish the independence of and , this implying that the law of coincides with the conditional law of B, given that . We provide the Karhunen-Loeve expansion on [0,1] of , making use of the Bessel functions J1/2 and J3/2. Applications and variants of these results are discussed. In particular, we establish a comparison theorem concerning the supremum distributions of and on [0,1].