Invariance principles for standard-normalized and self-normalized random fields

TL;DR

This paper explores invariance principles for set-indexed partial sums of stationary random fields, focusing on standard and self-normalization methods, with implications for understanding the asymptotic behavior of such fields.

Contribution

It introduces invariance principles for set-indexed partial sums of stationary fields under different normalization schemes, extending classical results to more complex index sets.

Findings

Invariance principles established for martingale-difference and independent fields.

Results applicable to both standard-normalized and self-normalized sums.

Provides theoretical foundations for asymptotic analysis of multi-dimensional random fields.

Abstract

We investigate the invariance principle for set-indexed partial sums of a stationary field $(X\_{k})\_{k\in\mathbb{Z}^{d}}$ of martingale-difference or independent random variables under standard-normalization or self-normalization respectively.