On possible mixing rates for some strong mixing conditions for N-tuplewise independent random fields

TL;DR

This paper demonstrates that for certain stationary random fields with N-tuplewise independence, the dependence coefficients under various strong mixing conditions can decay at any arbitrary rate, impacting limit theory applications.

Contribution

It constructs classes of random fields with N-tuplewise independence where dependence coefficients decay arbitrarily, advancing understanding of mixing conditions in high-dimensional random fields.

Findings

Dependence coefficients can decay arbitrarily fast for N-tuplewise independent fields.

Provides examples relevant to limit theorems involving strong mixing conditions.

Establishes related results on the structure of such random fields.

Abstract

For a given pair of positive integers $d$ and $N$ with $N \geq 2$, for strictly stationary random fields that are indexed by the $d$-dimensional integer lattice and satisfy $N$-tuplewise independence, the dependence coefficients associated with the $\rho$-, $\rho'$-, and $\rho^*$-mixing conditions can decay together at an arbitrary rate. Another, closely related result is also established. In particular, these constructions provide classes of examples pertinent to limit theory for random fields that involve such mixing conditions together with certain types of "extra" assumptions on the marginal and bivariate (or $N$-variate) distributions.