A central limit theorem for the rescaled Lévy area of two-dimensional fractional Brownian motion with Hurst index $H<1/4$

Abstract

Let $B=(B^{(1)},B^{(2)})$ be a two-dimensional fractional Brownian motion with Hurst index $\alpha \in (0,1/4)$. Using an analytic approximation $B(\eta )$ of $B$ introduced in \cite{Unt08}, we prove that the rescaled Lévy area process $(s,t)\to η^{\half(1-4α)}\int_s^t dB_{t_1}^{(1)}(η) \int_s^{t_1} dB_{t_2}^{(2)}(η)$ converges in law to $W_t-W_s$ where $W$ is a Brownian motion independent from $B$. The method relies on a very general scheme of analysis of singularities of analytic functions, applied to the moments of finite-dimensional distributions of the Lévy area.

70 pages, 1 figure

References in corpus (3)

• Central limit theorems for sequences of multiple stochastic integrals
• Renormalized self-intersection local time for fractional Brownian motion
• Stochastic calculus for fractional Brownian motion with Hurst exponent $H>1/4$: A rough path method by analytic extension

Cited by in corpus (1)

• Discretizing the fractional Levy area