Finite Sections of Random Jacobi Operators

Abstract

This article is about a problem in the numerical analysis of random operators. We study a version of the finite section method for the approximate solution of equations $Ax=b$ in infinitely many variables, where A is a random Jacobi (i.e., tridiagonal) operator. In other words, we approximately solve infinite second order difference equations with stochastic coefficients by reducing the infinite volume case to the (large) finite volume case via a particular truncation technique. For most of the paper we consider non-self-adjoint operators A, but we also comment on the self-adjoint case when simplifications occur.