Randomized limit theorems for stationary ergodic random processes and fields

TL;DR

This paper extends classical limit theorems like the CLT and invariance principle to ergodic homogeneous random fields using a randomized sampling approach, broadening their applicability.

Contribution

It generalizes key probabilistic limit theorems to all ergodic homogeneous random fields on ^m and ^m using a novel randomized method.

Findings

Extended the invariance principle to ergodic fields.

Generalized the Glivenko--Cantelli theorem for these fields.

Provided an improved CLT version for ergodic random fields.

Abstract

We consider "randomized" statistics constructed by using a finite number of observations a random field at randomly chosen points. We generalize the invariance principle (the functional CLT), the Glivenko--Cantelli theorem, the theorem about convergence to the Brownian bridge and the Kolmogorov theorem about the limit distribution of the empirical distribution function, as well as an improved version of the CLT in A. Tempelman, Randomized multivariate central limit theorems for ergodic homogeneous random fields, Stochastic Processes and their Applications. 143 (2022), 89-105. The randomized approach, introduced in the mentioned work, allows to extend these theorems to all ergodic homogeneous random fields on $\Z^m$ and $\R^m.$