Asymptotic properties of the derivative of self-intersection local time of fractional Brownian motion

Abstract

Let {Bt}t≥0 be a fractional Brownian motion with Hurst parameter 23<H<1. We prove that the approximation of the derivative of self-intersection local time, defined as αε=∫0T∫0tpε′(Bt−Bs)dsdt, where pε(x) is the heat kernel, satisfies a central limit theorem when renormalized by ε32−1H. We prove as well that for q≥2, the qth chaotic component of αε converges in L2 when 23<H<34, and satisfies a central limit theorem when renormalized by a multiplicative factor ε1−34H in the case 34<H<4q−34q−2.