Entropy and Grand Lebesgue Spaces approach for Prokhorov-Skorokhod continuity of random processes, with tail estimates

Abstract

We present in this paper a new sufficient condition for the so-called Prokhorov-Skorokhod continuity of random processes. Our conditions will be formulated in the terms of metric entropy generated by three-dimensional distribution of the considered random process (r.p.) in the parametric set, have a convenient and closed form, and generalize some previous results. We study also the conditions for weak compactness of the sequence of random processes in this space and as a consequence the Central Limit Theorem. Our consideration based on the theory of Prokhorov-Skorokhod spaces of random processes and on the theory of Banach spaces of random variables with exponential decreasing tails of distributions, namely, on the theory of Grand Lebesgue Spaces (GLS) of random variables.