Asymptotics of Karhunen-Lo{\`e}ve Eigenvalues for sub-fractional Brownian motion and its application

Abstract

In the present paper, the Karhunen-Lo{\`e}ve eigenvalues for a sub-fractional Brownian motion are considered in the case of $H>\frac12$. Rigorous large $n$ asymptotics for those eigenvalues are shown, based on functional analysis method. By virtue of these asymptotics, along with some standard large deviations results, asymptotically estimates for the closely related problem of small $L^2$-ball probabilities for a sub-fractional Brownian motion are derived. By the way, asymptotic analysis on the Karhunen-Lo{\`e}ve eigenvalues for the corresponding "derivative" process is also established.