A central limit theorem associated with sub-fractional Brownian motion and an application

Abstract

In this paper, we consider the asymptotic normality associated with the integral functionals $$\frac1{\eta(\varepsilon)} \int_0^T ( (S^H_{s+\varepsilon}-S^H_s )^2-\varepsilon^{2H} )ds, \varepsilon>0 $$ with $T>0$, where $\eta(\varepsilon)$ is an infinitesimal as $\varepsilon$ tends to zero. When $0＜H＜\frac34$ and $\eta(\varepsilon)=\varepsilon^{2H+\frac12}$, we show that there exists a standard Brownian motion $B$ such that it converges weakly to $B$ multiplied by a constant, and moreover when $H=\frac34$ and $\eta(\varepsilon)=\varepsilon^2\sqrt{-\log\varepsilon}$, we also show that there exists a standard Brownian motion $W$ such that it converges weakly to $\frac34W$. As an application we study the asymptotic normality of the estimator of parameter $\sigma>0$ in Ornstein-Uhlenbeck process $$ X^H_t=X^H_0+\sigma S^H_t-\beta\int_0^tX^H_sds, $$ by using the so-called generalized quadratic variation.