Conditioned random walks on linear groups I: construction of the target harmonic measure

TL;DR

This paper develops a new framework for conditioned random walks on linear groups, constructing a target harmonic measure and establishing foundational results towards a conditioned local limit theorem.

Contribution

It introduces the construction of the target harmonic measure and a novel approach using finite-size approximations for analyzing conditioned random walks.

Findings

Construction of the target harmonic measure.

Introduction of a reversed dual random walk with future dependence.

Reduction of complex walks to Markov chains with increasing dimensions.

Abstract

Our objective is to explore random walks on the general linear group, constrained to a specific domain, with a primary focus on establishing the conditioned local limit theorem. This paper marks the initial stride toward achieving this goal, specifically entailing the construction of a novel entity -- the target harmonic measure. This measure, together with the harmonic function, serves as a pivotal component in shaping the conditioned local limit theorem. Using a reversal identity, we introduce a reversed sequence characterized as a dual random walk with a perturbation depending on future observations. The investigation of such walks, which rely on future information, lies at the heart of this paper. To carry out this study, we develop an approach grounded in the finite-size approximation of perturbations, enabling us to simplify the investigation to an array of Markov chains with increasing dimensions.