A stochastic maximal inequality and related topics

As an alternative to the well-known methods of "chaining" and "bracketing" that have been developed in the study of random fields, a new stochastic maximal inequality is presented with a proof based on the formula for integration by parts. The main results of this paper are some central limit theorems in the space $\ell^\infty(\TT)$, the space of bounded functions on a set $ \TT $ equipped with the uniform metric, for some sequences of separable random fields of locally square-integrable martingales with the help also of entropy methods. As special cases, some new results for i.i.d.\ random sequences, including an Donsker theorem and a maximal inequality for empirical processes indexed by classes of sets or functions, are obtained. Some other topics of independent interest, such as seeking a sufficient condition for the existence of bounded continuous version of given separable, centered Gaussian random field and doing that for the VC-dimension of given countable class of sets to be finite, are also discussed.