Central Limit Theorem, Weak Law of Large Numbers for Martingales in Banach Spaces, and Weak Invariance Principle - A Quantitative Study

Abstract

This article deals with quantitative results by involving the standard modulus of continuity in Banach spaces. These concern convergence in distribution for Banach space-valued martingale difference sequences and weak convergence of the distribution of random polygonal lines to the Wiener-measure on C([0, 1]). A general theorem is given with applications to the central limit theorem and weak law of large numbers for Banach space-valued martingales. Another general theorem is presented on the weak invariance principle with an application to a central limit theorem for real-valued martingales. The exposed results generalize earlier related results of Butzer, Hahn, Kirschfink, and Roeckerath.