Karhunen–Loeve expansions for the detrended Brownian motion

The detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the subspace spanned by linear functions. Karhunen–Loeve expansion for the process is obtained, together with the explicit formula for the Laplace transform of the squared L2L2 norm. Distribution identities are established in connection with the second order Brownian bridge developed by MacNeill (1978). As applications, large and small deviation asymptotic behaviors for the L2L2 norm are given.