On the quenched functional CLT in 2d random sceneries, examples

TL;DR

This paper establishes a quenched functional CLT for sums of random fields along 2D random walks in various scenarios, including iid fields, automorphism-generated fields, and Lorentz processes, advancing understanding of stochastic processes in random environments.

Contribution

It proves a quenched FCLT in 2D for different types of random fields and processes, including iid, automorphism-generated, and Lorentz processes, extending previous results in the field.

Findings

Quenched FCLT holds for iid random sceneries with second order moments.

Quenched FCLT applies to fields generated by automorphisms of a torus.

Results include a quenched FCLT for Lorentz processes in random sceneries.

Abstract

We prove a quenched functional central limit theorem (quenched FCLT) for the sums of a random field (r.f.) along a 2d-random walk in different situations: when the r.f. is iid with a second order moment (random sceneries), or when it is generated by the action of commuting automorphisms of a torus. We consider also a quenched version of the FCLT when the random walk is replaced by a Lorentz process in the random scenery.