Karhunen-Loeve expansions for the m-th order detrended Brownian motion

The m-th order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion onto the subspace spanned by polynomials of degree up to m. We obtain the Karhunen-Loeve expansion for the process and establish a connection with the generalized (m-th order) Brownian bridge developed by MacNeill (1978) in the study of distributions of polynomial regression. The resulting distribution identity is also verified by a stochastic Fubini approach. As applications, large and small deviation asymptotic behaviors for the L ₂ norm are given.