CHAPTER 8 - Convergence and Limit Theorems for Random Polynomials

Abstract

Convergence and limit theorems for random variables, random functions, and probability measures are of fundamental importance in probability theory and its applications. In classical probability theory, limit theorems are primarily concerned with the limiting behavior of sums of random variables however, in modern probabilistic analysis, the limit properties of products of random variables and random operators have been investigated. This chapter highlights the limiting behavior of the number of zeros of random algebraic and random trigonometric polynomials. It also presents some results on the averaging problem for the zeros of random algebraic polynomials, limit theorems for products of random algebraic polynomials, and random companion matrices. The chapter discusses theorems that are concerned with (1) the limiting behavior of the random measures Nn(,B, ω)—the number of zeros of random algebraic and trigonometric polynomials that are contained in a Borel set B of the complex plane Z, (2) the limiting behavior of the zeros of random algebraic polynomials, and (3) the limiting behavior of products of random algebraic polynomials and of random companion matrices and sums of random companion matrices.