ABSTRACT
In this chapter are presented quantitative results which involve the standard modulus of continuity in Banach spaces. These give the convergence in distribution for Banach space-valued martingale difference sequences and the weak convergence of the distributions of random polygonal lines to the Wiener-measure on C([0,1]). A general theorem is presented with applications to the central limit theorem and weak law of large numbers for Banach space-valued martingales. Also, a general theorem is given on the weak invariance principle with an application to a central limit theorem for real-valued martingales. This treatment is based on [16], [55] and [56]. In particular, the exhibited main inequalities here involve the standard modulus of continuity of certain Fréchet derivative of the acting function in the associated weak convergences.
