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

TL;DR

This paper derives large n asymptotics for the eigenvalues of the Karhunen-Loève expansion of sub-fractional Brownian motion with H>1/2, and applies these results to estimate small ball probabilities and analyze related derivative processes.

Contribution

It provides the first rigorous asymptotic analysis of eigenvalues for sub-fractional Brownian motion in the H>1/2 case, with applications to small ball probability estimates.

Findings

Asymptotic formulas for eigenvalues are established.

Small L^2-ball probabilities are asymptotically estimated.

Eigenvalue analysis for the derivative process is also provided.

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.