A chaos hypothesis for some large systems of random equations

Abstract

This paper is about the behavior of solutions to large systems of linear algebraic and differential equations when the coefficients are random variables. We will prove a law of large numbers and a central limit theorem for the solutions of certain algebraic systems, and the weak convergence to a Gaussian process for the solution of a system of differential equations. Some of the results were surprisingly difficult to prove, but they are all easily anticipated from a "chaos hypothesis": i.e. an assumption of near independence for the components of the solutions of large systems of weakly coupled equations. © 1982 Springer-Verlag.