Let $\{ B_{ t}\}_{ t\geq0}$ be a fractional Brownian motion with Hurst parameter $\frac{ 2}{ 3}<H<1$. We prove that the approximation of the derivative of self-intersection local time, defined as \begin{ align*} \alpha_{ \varepsilon} &= \int_{ 0}^{ T}\int_{ 0}^{ t}p'_{ \varepsilon}(B_{ t}-B_{ s})\text{ d}s\text{ d}t, \end{ align*} where $p_\varepsilon(x)$ is the heat kernel, satisfies a central limit theorem when renormalized by $\varepsilon^{ \frac{ 3}{ 2}-\frac{ 1}{ H}}$. We prove as well that for $q\geq2$, the $q$-th chaotic component of $\alpha_{ \varepsilon}$ converges in $L^{ 2}$ when $\frac{ 2}{ 3}<H<\frac{ 3}{ 4}$, and satisfies a central limit theorem when renormalized by a multiplicative factor $\varepsilon^{ 1-\frac{ 3}{ 4H}}$ in the case $\frac{ 3}{ 4}<H<\frac{ 4q-3}{ 4q-2}$.
Categories: math.PR
