Moment and tail estimates and Banach space valued Non-Central Limit Theorem (NCLT) for sums of multi-indexed random variables, processes and fields

TL;DR

This paper establishes moment and tail estimates, along with conditions for the Non-Central Limit Theorem, for normalized sums of multi-indexed random variables, extending to processes and fields with applications to V-statistics.

Contribution

It introduces new moment and tail bounds and provides conditions for NCLT in multi-dimensional settings, utilizing degenerate approximation and Grand Lebesgue Spaces.

Findings

Derived explicit moment and tail estimates for multi-indexed sums.

Established sufficient conditions for NCLT in various function spaces.

Provided examples demonstrating the sharpness of the estimates.

Abstract

We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as the theory of Grand Lebesgue Spaces (GLS) of measurable functions (random variables).