Let \{ B_t^H,t\geq0\} be a d-dimensional fractional Brownian motion. We prove that the approximation of the first-order derivative of self-intersection local time, defined as \alpha_{ \varepsilon,t}^{ (1)}(0)=-\int_0^t\int_0^sp_\varepsilon^{ (1)}(B_s^H-B_r^H)\d r\d s, where p_\varepsilon^{ (1)}(x_1,\cdots,x_d):=\partial _{ x_1}p(x_1,\cdots,x_d) and p_\varepsilon(x)=(2\pi\varepsilon)^{ -d/2}e^{ |x|^2/2\varepsilon},x\in\mathbb{ R}^d, d\geq2 is the heat kernel, exits in L^2 sense if and only if H<\frac{ 3}{ 2(1+d)} and satisfies three different central limit theorems when normalized by \varepsilon^{ \frac d2+1-\frac1H} for H>\frac12 and d\geq2, normalized by \varepsilon^{ \frac d2+\frac12-\frac 3{ 4H}} for \frac{ 3}{ 2(1+d)}<H<\frac12 and d\geq3, and normalized by \log(1/\varepsilon)^{ -\frac12} for the critical case H=\frac{ 3}{ 2(1+d)} and d\geq3.
Categories: math.PR