半鞅序列积分误差的极限过程的收敛定理

Jacod, Jakubowski和M\'emin讨论了与单个独立增量过程$X$的误差过程$^n\!X =X_t-X_{nt]/n}$相关的积分误差过程$Y^n(X)$和$Z^{n,p}(X)$, 研究了半鞅序列$\{(nY^n(X),nZ^{n,p}(X))\}_{n\ge 1}$的极限定理. 记半鞅序列$\{(nY^n(X),nZ^{n,p}(X))\}_{n\ge1}$的极限过程为$(Y(X),Z^p(X))$, Jacod等给出了其极限过程$(Y(X)$, $Z^p(X))$的表达式. 本文将研究半鞅序列$\{X^n\}_{n\ge1}$积分误差的极限过程$Y(X^n)$和$Z^{p}(X^n)$的收敛定理, 主要研究半鞅序列$\{(X^n,Y(X^n),Z^p(X^n))\}_{n\ge1}$的依分布弱收敛和依分布稳定收敛.

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

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