Abstract
Based on the norm in the Hilbert SpaceL2[0,1], the second order detrended Brownian motion is defined as the orthogonal component of projection of the standard Brownian motion into the space spanned by nonlinear function subspace. Karhunen-Loève expansion for this process is obtained together with the relationship of that of a generalized Brownian bridge. As applications, Laplace transform, large deviation, and small deviation are given.
Highlights
Let X = {X(t), 0 ≤ t ≤ 1} be a centered and continuous Gaussian process on [0, 1] with covariance functionKX (t, s) = EX (t) X (s) . (1)The Karhunen-Loeve expansion of X is given by the series ∞X (t) = ∑ ηk√λkfk (t), (2)k=1 where {ηk, k ≥ 1} is a sequence of i.i.d
Deheuvels et al in [1,2,3,4] provided the Karhunen-Loeve expansions for the processes that are related with Brownian motion
The Karhunen-Loeve expansion for detrended Brownian motion has been studied by Ai et al [5]
Summary
KarhunenLoeve expansion for this process is obtained together with the relationship of that of a generalized Brownian bridge. K=1 where {ηk, k ≥ 1} is a sequence of i.i.d. N(0, 1) random variables and {λk, k ≥ 1} is at most the countable set of eigenvalues of Fredholm integral operator Deheuvels et al in [1,2,3,4] provided the Karhunen-Loeve expansions for the processes that are related with Brownian motion. The Karhunen-Loeve expansion for detrended Brownian motion has been studied by Ai et al [5].
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
