Asymptotics of Karhunen–Loève Eigenvalues for Sub-Fractional Brownian Motion and Its Application

Abstract

In the present paper, the Karhunen–Loève eigenvalues for a sub-fractional Brownian motion are considered. Rigorous large n asymptotics for those eigenvalues are shown, based on the functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotical estimates for the small L2-ball probabilities for a sub-fractional Brownian motion are derived. Asymptotic analysis on the Karhunen–Loève eigenvalues for the corresponding “derivative” process is also established.