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Universality of Persistence of Random Polynomials

Promit Ghosal, Sumit Mukherjee

Abstract

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree $2n$ with coefficients given by an i.i.d. sequence of mean-zero, variance-1 random variables, multiplied by an $\fracα{2}$-regularly varying sequence for $α>-1$. We show that the probability of no real zeros is asymptotically $n^{-2(b_α+b_0)}$, where $b_α$ is the persistence exponents of a mean-zero, one-dimensional stationary Gaussian processes with covariance function as $\mathrm{sech}((t-s)/2)^{α+1}$. Our work generalizes the previous results of Dembo et al. [DPSZ02] and Dembo \& Mukherjee [DM15] by removing the requirement of finite moments of all order or Gaussianity. In particular, in the special case $α= 0$, our findings confirm a conjecture by Poonen and Stoll [PS99, Section 9.1] concerning random polynomials with i.i.d. coefficients.

Universality of Persistence of Random Polynomials

Abstract

We investigate the probability that a random polynomial with independent, mean-zero and finite variance coefficients has no real zeros. Specifically, we consider a random polynomial of degree with coefficients given by an i.i.d. sequence of mean-zero, variance-1 random variables, multiplied by an -regularly varying sequence for . We show that the probability of no real zeros is asymptotically , where is the persistence exponents of a mean-zero, one-dimensional stationary Gaussian processes with covariance function as . Our work generalizes the previous results of Dembo et al. [DPSZ02] and Dembo \& Mukherjee [DM15] by removing the requirement of finite moments of all order or Gaussianity. In particular, in the special case , our findings confirm a conjecture by Poonen and Stoll [PS99, Section 9.1] concerning random polynomials with i.i.d. coefficients.

Paper Structure

This paper contains 24 sections, 30 theorems, 266 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof Ideas
  3. Upper bound
  4. Fixation of notations
  5. Proof of $\mathfrak{LimSup}$
  6. Proof of Lemma \ref{['lem:clm']}
  7. Proof of Lemma \ref{['lem:weak']}
  8. Proof of Lemma \ref{['lem:KextBound']}
  9. Lower Bound
  10. Fixation of Notations
  11. Proof of $\mathfrak{LimInf}$
  12. Proof of Proposition \ref{['prop:r=-1']}: $r=- 1$ Case
  13. Lower Bound on $\mathds{P}(B_{1p})$
  14. Upper Bound on $\mathds{P}(\neg B_{2p})$
  15. Upper Bound on $\mathds{P}(\neg B_{3p})$ & $\mathds{P}(\neg B_{4p})$
  16. ...and 9 more sections

Key Result

Theorem 1.1

Fix $\alpha>-1$, and assume $a_i=\sqrt{R(i)}\xi_i$ with $\{\xi_i\}_{0\le i\le n}$ i.i.d. random variables with $\mathbb{E}[\xi_i]=0$ and $\mathrm{Var}(\xi_i) =1$. Then we have where $b_{\alpha}$ is defined via no-zero crossing probability of a stationary Gaussian process $\{Y^{(\alpha)}_t\}_{t\geq 0}$, i.e., where $(Y^{(\alpha)}_t)_{t\geq 0}$ is a mean zero stationary Gaussian process with the f

Figures (2)

  • Figure 1: Illustration of contributing intervals in $\mathfrak{LimSup}$ Case. The probability of no real zero of $Q_n(x)$ on the intervals $\mathfrak{D}_1$ or $\mathfrak{D}_3$ is $O(n^{-b_{\alpha}})$. Similarly, the probability of no real zero of $Q_n(x)$ on the intervals $\mathfrak{D}_2$ or $\mathfrak{D}_4$ is $O(n^{-b_{0}})$. Combining these bounds with 'near' independence of these four events shows $p^{(\alpha)}_{2n}= O(n^{-2(b_{\alpha}+b_0)})$.
  • Figure 2: Illustration of how $\mathbb{R}$ is divided into intervals in the $\mathfrak{LimInf}$ Case. Indeed, $[-\mathrm{e}^{\frac{h}{\log n}}, -\mathrm{e}^{\frac{K}{n}}]$, $[-\mathrm{e}^{\frac{-K}{n}}, -\mathrm{e}^{-\frac{h}{\log n}}]$, $[\mathrm{e}^{-\frac{h}{\log n}}, \mathrm{e}^{\frac{-K}{n}}]$ and $[\mathrm{e}^{\frac{-K}{n}}, \mathrm{e}^{\frac{h}{\log n}}]$ are the intervals which mainly contribute to the lower bound of the probability of no real zeros of $Q_{2n}(x)$ and they play same role as in the intervals $\mathfrak{D}_4, \mathfrak{D}_3, \mathfrak{D}_1$ and $\mathfrak{D}_2$ respectively as in the proof of $\mathfrak{LimSup}$.

Theorems & Definitions (56)

  • Theorem 1.1
  • Remark 1.1
  • Lemma 3.1
  • Lemma 3.2
  • Definition 3.3
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['lem:clm']}
  • Lemma 3.5
  • proof : Proof of Lemma \ref{['lem:weak']}
  • proof : Proof of Lemma \ref{['lem:weak*']}
  • ...and 46 more