Functional limit theorem for the self-intersection local time of the fractional Brownian motion

Abstract

Let $\{B_t{\}}_{t\ge 0}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $0<H<1$, where $d\ge 2$. Consider the approximation of the self-intersection local time of $B$, defined as \begin{align*} I_{T}^{\varepsilon} &=\int_{0}^{T}\int_{0}^{t}p_{\varepsilon}(B_{t}-B_{s})dsdt, \end{align*} where $p_{\epsilon }(x)$ is the heat kernel. We prove that the process $\{I_T^{\epsilon }-\mathbb{E}\left[I_T^{\epsilon }\right]{\}}_{T\ge 0}$, rescaled by a suitable normalization, converges in law to a constant multiple of a standard Brownian motion for $\frac{3}{2d}<H\le \frac{3}{4}$ and to a multiple of a sum of independent Hermite processes for $\frac{3}{4}<H<1$, in the space $C[0,\infty )$, endowed with the topology of uniform convergence on compacts.

References in corpus (2)

• Central limit theorems for sequences of multiple stochastic integrals
• Renormalized self-intersection local time for fractional Brownian motion