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February 2015 No zero-crossings for random polynomials and the heat equation
Amir Dembo, Sumit Mukherjee
Ann. Probab. 43(1): 85-118 (February 2015). DOI: 10.1214/13-AOP852

Abstract

Consider random polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ of independent mean-zero normal coefficients $a_{i}$, whose variance is a regularly varying function (in $i$) of order $\alpha$. We derive general criteria for continuity of persistence exponents for centered Gaussian processes, and use these to show that such polynomial has no roots in $[0,1]$ with probability $n^{-b_{\alpha}+o(1)}$, and no roots in $(1,\infty)$ with probability $n^{-b_{0}+o(1)}$, hence for $n$ even, it has no real roots with probability $n^{-2b_{\alpha}-2b_{0}+o(1)}$. Here, $b_{\alpha}=0$ when $\alpha\le-1$ and otherwise $b_{\alpha}\in(0,\infty)$ is independent of the detailed regularly varying variance function and corresponds to persistence probabilities for an explicit stationary Gaussian process of smooth sample path. Further, making precise the solution $\phi_{d}({\mathbf{x}},t)$ to the $d$-dimensional heat equation initiated by a Gaussian white noise $\phi_{d}({\mathbf{x}},0)$, we confirm that the probability of $\phi_{d}({\mathbf{x}},t)\neq0$ for all $t\in[1,T]$, is $T^{-b_{\alpha}+o(1)}$, for $\alpha=d/2-1$.

Citation

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Amir Dembo. Sumit Mukherjee. "No zero-crossings for random polynomials and the heat equation." Ann. Probab. 43 (1) 85 - 118, February 2015. https://doi.org/10.1214/13-AOP852

Information

Published: February 2015
First available in Project Euclid: 12 November 2014

zbMATH: 1312.60036
MathSciNet: MR3298469
Digital Object Identifier: 10.1214/13-AOP852

Subjects:
Primary: 26C10, 60G15
Secondary: 26A12, 35K05

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 1 • February 2015
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