Asymptotic properties for q-th chaotic component of derivative of self-intersection local time of fractional Brownian motion

Abstract

Let {BtH}t≥0 be a fractional Brownian motion with Hurst parameter H∈(0,1). Consider the approximation of derivative of self-intersection local time of BH, defined asαε=∫0T∫0tpε′(BtH−BsH)dsdt, where pε(x) is the heat kernel. For q≥2, renormalized by different factors, we prove two different central limit theorems for q-th chaotic component of αε when H=34 and H=4q−34q−2, respectively. The results are obtained based on the fourth moment theorem introduced in [8].