Stochastic fields with paths in arbitrary rearrangement invariant spaces

TL;DR

This paper establishes conditions under which paths of a random process belong to certain rearrangement invariant spaces and satisfy the CLT, with potential applications in probability theory.

Contribution

It provides new sufficient conditions for path inclusion in r.i. spaces and for the CLT to hold in these spaces, expanding understanding of stochastic process behavior.

Findings

Paths of random processes can belong to specific r.i. spaces under certain conditions.

The CLT can be established in these r.i. spaces for the paths of the processes.

Applications demonstrate practical relevance of the theoretical results.

Abstract

We obtain sufficient conditions for belonging of almost all paths of a random process to some fixed rearrangement invariant (r.i.) Banach functional space, and to satisfying the Central Limit Theorem (CLT) in this space. We describe also some possible applications.