次分数Brown运动一个积分泛函的中心极限定理及其应用  

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

作　　者：孙西超 闫理坦[2] SUN XiChao YAN LiTan

机构地区：[1]蚌埠学院理学院数学系,蚌埠233010 [2]东华大学理学院数学系,上海201620

出　　处：《中国科学：数学》2017年第9期1055-1076,共22页Scientia Sinica：Mathematica

基　　金：国家自然科学基金(批准号:11571071;11626033和11426036);安徽省自然科学重点项目(批准号:KJ2016A453和KJ2017A568);安徽省优秀人才支持计划重点项目(批准号:gxyq ZD2016354)资助项目

摘　　要：假设S^H是Hurst参数为0<H<1的次分数Brown运动.本文研究积分过程1/(η(ε))∫_0~T((S_(s+ε)~H-S_s^H)~2-ε^(2H))ds,ε>0的渐近分布,其中T>0,η(ε)表示一个当ε→0时的无穷小量.当0<H<3/4和η(ε)=ε^(2H+1/2时,本文证明了上述积分弱收敛于一个标准Brown运动B的常数倍;当H=3/4和η(ε)=ε(-logε)~1/2时,证明了存在另一标准Brown运动W,使得上述积分弱收敛于3/4W.为应用,本文利用广义二次变差建立了Ornstein-Uhlenbeck(O-U)过程X_t^H=X_0~H+ σS_t^H-β∫_0~tX_s^Hds,中参数σ>0的估计量,并给出其渐近正态性.In this paper, we consider the asymptotic normality associated with the integral functionals 1/η（ε）∫0^T（（Ss＋ε^H-Ss^H）^2-ε^2H）ds,ε〉03 and η（ε）= ε^2H＋1/2, we show with T 〉 0, where η（ε） is an infinitesimal as e tends to zero. When 0 〈 H 〈 that there exists a standard Brownian motion B such that it converges weakly to B multiplied by a constant, 3 and η（ε）=ε^2H＋1/2, we also show that there exists a standard Brownian motion and moreover when H = 3 W such that it converges weakly to 3/4W. As an application we study the asymptotic normality of the estimator of parameter a 〉 0 in Ornstein-Uhlenbeck processby using the so-called generalized quadratic variation.

关 键 词：次分数Brown运动 中心极限定理 O-U过程 广义二次变差 

分 类 号：O211.4[理学—概率论与数理统计]