Quenched invariance principles for orthomartingale-like sequences

TL;DR

This paper establishes quenched invariance principles for orthomartingale-like sequences, extending CLT results to non-equilibrium initial conditions and applying these to linear and nonlinear random fields.

Contribution

It introduces quenched invariance principles for orthomartingales and extends these results to broader classes of random fields through approximation and decomposition techniques.

Findings

Orthomartingales can satisfy CLT without quenched invariance.

Constructed example shows CLT does not imply quenched CLT.

Results apply to both linear and nonlinear random fields.

Abstract

In this paper we study the central limit theorem and its functional form for random fields which are not started from their equilibrium, but rather under the measure conditioned by the past sigma field. The initial class considered is that of orthomartingales and then the result is extended to a more general class of random fields by approximating them, in some sense, with an orthomartingale. We construct an example which shows that there are orthomartingales which satisfy the CLT but not its quenched form. This example also clarifies the optimality of the moment conditions used for the validity of our results. Finally, by using the so called orthomartingale-coboundary decomposition, we apply our results to linear and nonlinear random fields.