A Probabilistic Version of Eneström–Kakeya Theorem for Certain Random Polynomials

Abstract

This paper presents a comprehensive exploration of a probabilistic adaptation of the Eneström–Kakeya theorem, applied to random polynomials featuring various coefficient distributions. Unlike the deterministic rendition of the theorem, our study dispenses with the necessity of any specific coefficient order. Instead, we consider coefficients drawn from a spectrum of sets with diverse probability distributions, encompassing finite, countable, and uncountable sets. Furthermore, we provide a result concerning the probability of failure of Schur stability for a random polynomial with coefficients distributed independently and identically as standard normal variates. We also provide simulations to corroborate our results.

Keywords

• Eneström–Kakeya theorem
• random polynomials
• normal distribution
• Schur stability