On a multidimensional martingale with given conditional covariance structure (Q1917620)

Let \(M^n\) be a sequence of discrete time \(\mathbb{R}^m\) valued martingales with the conditional covariance matrix \(\Gamma^n_k \dot = \biggl \{E^{F^n_{k - 1}} \bigl( M^n_k (j) M^n_k (i) \bigr) \biggr\}_{i,j = 1, \dots, m}\), here \(M^n_k (j)\) is the \(j\)th component of the vector \(M^n_k\). Define the corresponding matrix for the increment \(\Delta M^n_k\) and denote it by \(Q^n_k\). The first results of the paper consider the approximation of the positive definite matrix \(Q\) with the help of the matrices \(Q^n_k\) and the approximation of the martingale sequence \(M^n\) with another martingale sequence \(\widehat M^n\) which has covariance matrix equal to \(Q\) and the jumps of \(\widehat M^n\) can be controlled with the difference \(|Q - \sum_k Q^n_k |\) and the jumps of the original martingale sequence \(M^n\). The approximation results are used to obtain some rates of convergence results for vector valued martingales, which are quite natural in this context.