2017/07/27 by Yoichi Nishiyama, Nishiyama, Yoichi
Mathematics · #Advanced Banach Space Theory
paper · pdf · doi:10.48550/arxiv.1707.08829
As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new method, which is based on a \em stochastic maximal inequality derived by using the formula for integration by parts and on a new concept named \em strict countability, is presented. The main results are some weakconvergence theorems for sequences of separable random fields of discrete-time martingales under the uniform topology with the help also of entropy methods. As special cases, some new results for i.i.d. random sequences, including a new Donsker theorem and a moment bound for suprema of empirical processes indexed by classes of sets or functions, are obtained.