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

The processes of the form y(K)(t) = B(t) - 6Kt(1 - t) integral(1)(0) B(u)du, where K is a constant, and B(center dot) a Brownian bridge, are investigated. We show that y(0)(center dot) and y(2)(center dot) are both Brownian bridges, and establish the independence of and integral(1)(0) B(u)du, this implying that the law of y(1)(center dot) coincides with the conditional law of B, given that integral B-1(0)(u) du = 0. We provide the Karhunen-Loeve expansion on [0, 1] of y(1)(center dot), making use of the Bessel functions J(1/2) and J(3/2). Applications and variants of these results are discussed. In particular, we establish a comparison theorem concerning the supremum distributions of y(K')(center dot) and y(K'')(center dot)) on [0, 1]. (c) 2007 Elsevier B.V. All rights reserved.