NOTE ON WEAK CONVERGENCE, LARGE DEVIATIONS, AND THE BOUNDED APPROXIMATION PROPERTY.

Given a Banach space (E, ∥ · ∥ ) having the bounded approximation property, we provide a paradigm that can be used to establish large deviation principles for sequences of random elements (Xn)n≧1 taking values in E. A similar paradigm is given to establish weak convergence of (Xn)n≧1 to a tight Borel measure in (E,∥ · ∥). We then make use of these tools to establish a functional limit law of the iterated logarithm for the smoothed empirical process in Hölder topology. [ABSTRACT FROM AUTHOR]