Uniform Central Limit Theorem for martingales

TL;DR

This paper establishes sufficient conditions for the Central Limit Theorem to hold for martingale difference sequences in Banach spaces of continuous functions, utilizing entropy and Grand Lebesgue Spaces theory.

Contribution

It introduces new sufficient conditions for the CLT in Banach spaces for martingales, combining entropy methods and Grand Lebesgue Spaces theory.

Findings

Established CLT conditions in Banach spaces for martingales

Connected entropy conditions with tail behavior of distributions

Applied Grand Lebesgue Spaces to analyze tail decay

Abstract

We study some sufficient conditions imposed on the sequence of martingale differences (m.d.) in the separable Banach spaces of continuous functions defined on the metric compact set for the Central Limit Theorem in this space. We taking into account the classical entropy terms, and use the theory of the so-called Grand Lebesgue Spaces of random variables having power and exponential decreasing tail of distribution.