A large sample test for the length of memory of stationary symmetric stable random fields via nonsingular $\mathbb{Z}^d$-actions

TL;DR

This paper introduces a large sample statistical test for determining the memory length of stationary symmetric alpha-stable random fields, leveraging ergodic theory and block maxima ratios.

Contribution

It develops a novel large sample test based on block maxima ratios, utilizing ergodic theory of nonsingular 5^d-actions to assess memory length in stable random fields.

Findings

Test power approaches one with increasing sample size under longer memory alternatives.

The method effectively distinguishes different memory lengths in stationary symmetric 5-stable fields.

Ergodic theory is crucial for the test's design and analysis.

Abstract

Based on the ratio of two block maxima, we propose a large sample test for the length of memory of a stationary symmetric $\alpha$-stable discrete parameter random field. We show that the power function converges to one as the sample-size increases to infinity under various classes of alternatives having longer memory in the sense of Samorodnitsky(2004). Ergodic theory of nonsingular $\mathbb{Z}^d$-actions play a very important role in the design and analysis of our large sample test.