A general Kolmogorov-Chentsov type theorem on general metric spaces with applications to limit theorems for Banach-valued processes

TL;DR

This paper extends the Kolmogorov-Chentsov theorem to general metric spaces, providing a universal criterion for the regularity of Banach-valued stochastic processes and establishing new limit theorems.

Contribution

It introduces a generalized modulus of continuity theorem applicable to arbitrary metric spaces, enabling new limit theorems for Banach-valued processes.

Findings

Provides conditions for H"older-continuous modifications in general metric spaces

Establishes a criterion for uniform tightness of processes with continuous paths

Derives central limit theorems for Banach-valued processes

Abstract

The paper deals with moduli of continuity for paths of random processes indexed by a general metric space $\Theta$ with values in a general metric space $\mathcal{X}$. Adapting the moment condition on the increments from the classical Kolmogorov-Chentsov theorem, the obtained result on the modulus of continuity allows for H\"older-continuous modifications if the metric space $\mathcal{X}$ is complete. This result is universal in the sense that its applicability depends only on the geometry of the space $\Theta$. In particular, it is always applicable if $\Theta$ is a bounded subset of a Euclidean space or a relatively compact subset of a connected Riemannian manifold. The derivation is based on refined chaining techniques developed by Talagrand. As a consequence of the main result a criterion is presented to guarantee uniform tightness of random processes with continuous paths.This is applied to find central limit theorems for Banach-valued random processes.