A strong law of large numbers for real roots of random polynomials
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866929288525119488 |
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| author | Do, Yen Q. |
| author_facet | Do, Yen Q. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_06353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A strong law of large numbers for real roots of random polynomials Do, Yen Q. Probability 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. |
| title | A strong law of large numbers for real roots of random polynomials |
| topic | Probability |
| url | https://arxiv.org/abs/2403.06353 |