Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A strong law of large numbers for real roots of random polynomials

Yen Q. Do

Abstract

We consider random polynomials $p_n(x)=ξ_0+ξ_1+\dots+ξ_n x^n$ whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded $(2+ε)^{th}$ moment (for some $ε>0$), also known as the Kac polynomials. Let $N_n$ denote the number of real roots of $p_n$. In this paper, motivated by a question from Igor Pritsker, we prove that almost surely the following convergence holds: \begin{eqnarray*} \lim_{n\to\infty} \frac{N_n([-1,1])}{\log n} &=& \frac 1 π. \end{eqnarray*} This convergence could be viewed as a local strong law for the real roots. The main ingredient in the proof is a set of maximal inequalities that reduces the proof to proving convergence along lacunary subsequences, which in turn follows from a recent concentration estimate of Can--Nguyen.

A strong law of large numbers for real roots of random polynomials

Abstract

We consider random polynomials whose coefficients are independent and identically distributed with zero mean, unit variance, and bounded moment (for some ), also known as the Kac polynomials. Let denote the number of real roots of . In this paper, motivated by a question from Igor Pritsker, we prove that almost surely the following convergence holds: \begin{eqnarray*} \lim_{n\to\infty} \frac{N_n([-1,1])}{\log n} &=& \frac 1 π. \end{eqnarray*} This convergence could be viewed as a local strong law for the real roots. The main ingredient in the proof is a set of maximal inequalities that reduces the proof to proving convergence along lacunary subsequences, which in turn follows from a recent concentration estimate of Can--Nguyen.
Paper Structure (20 sections, 21 theorems, 148 equations)

This paper contains 20 sections, 21 theorems, 148 equations.

Table of Contents

  1. Introduction
  2. Strategy
  3. Notations and convention
  4. Outline of the proof of Theorem \ref{['t.main']}
  5. The lacunary case
  6. A small ball inequality for random polynomials
  7. Proof of Theorem \ref{['t.smallBE']}, step 1: a characteristic function bound
  8. Proof of Theorem \ref{['t.smallBE']}, step 2: proof of the small ball estimates
  9. Maximal estimates and convergence for small roots
  10. From maximal estimates to almost sure convergence: Proof of Lemma \ref{['l.near0-conv-alt']} using Lemma \ref{['l.near0-max']}
  11. Proof of Lemma \ref{['l.near0-max']}
  12. Maximal estimates and convergence near $1$
  13. An auxiliary estimate
  14. Proof of Lemma \ref{['l.near1-max']}
  15. Proof of Lemma \ref{['l.near1-max-sub']}
  16. ...and 5 more sections

Key Result

Theorem 1.1

Assume that $(\xi_j)_{j\ge 0}$ are iid with zero mean, unit variance, and bounded $(2+\epsilon)^{th}$ moment for some $\epsilon>0$. Then almost surely the following convergence holds: Furthermore, analogous results hold for $N_n[0,1]$ and $N_n[-1,0]$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Lemma 4.2
  • ...and 14 more