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.