半鞅序列积分误差的极限过程的收敛定理

Convergence Theorems of the Limit Processes of Integrated Errors of Semimartingale Sequence

Jacod,Jakubowski和Mémin讨论了与单个独立增量过程X的误差过程nX=Xt-X[nt]/n相关的积分误差过程Yn(X)和Zn,p(X),研究了半鞅序列{(nYn(X),nZn,p(X))}n≥1的极限定理.记半鞅序列{(nYn(X),nZn,p(X))}n≥1的极限过程为(Y(X),Zp(X)),Jacod等给出了其极限过程(Y(X),Zp(X))的表达式.本文将研究半鞅序列{Xn}n≥1积分误差的极限过程Y(Xn)和Zp(Xn)的收敛定理,主要研究半鞅序列{(Xn,Y(Xn),Zp(Xn))}n≥1的依分布弱收敛和依分布稳定收敛.

Jacod,Jakubowski and M'emin studied the integrated error processes Y n(X) and Zn,p(X) which relates to the error process nXt = Xt-X[nt]/n for semimartingale X with independent increments.And they also investigated the limit theorems for the semimartingale sequence {(Y(Xn),Zp(Xn))}n≥1.If denote the limit points of {(Y(Xn),Zp(Xn))}n≥1 by(Y(X),Zp(X)),Jacod et al.gave the formula of(Y(X),Zp(X)).In this paper,we will investigate the convergence theorems of Y(Xn) and Zp(Xn) for semimartingale sequence {Xn}n≥1.We study mainly the convergence in law and the stable convergence in law of {(Xn,Y(Xn),Zp(Xn))}n≥1.