The Common Limit of the Linear Statistics of Zeros of Random Polynomials and Their Derivatives

Abstract

Let $p_n(x)$ be a random polynomial of degree $n$ and $\{Z_j^{(n)}{\}}_{j=1}^n$ and $\{X_j^{n,k}{\}}_{j=1}^{n-k},k<n$, be the zeros of $p_n$ and $p_n^{(k)}$, the $k$th derivative of $p_n$, respectively. We show that if the linear statistics $\frac{1}{a_n}\left[f\left(\frac{Z_1^{(n)}}{b_n}\right)+\cdots +f\left(\frac{Z_n^{(n)}}{b_n}\right)\right]$ associated with $\{Z_j^{(n)}\}$ has a limit as $n\to \infty$ at some mode of convergence, the linear statistics associated with $\{X_j^{n,k}\}$ converges to the same limit at the same mode. Similar statement also holds for the centered linear statistics associated with the zeros of $p_n$ and $p_n^{(k)}$, provided the zeros $\{Z_j^{(n)}\}$ and the sequences $\{a_n\}$ and $\{b_n\}$ of positive numbers satisfy some mild conditions.

References in corpus (3)

• Central limit theorem for linear eigenvalue statistics of random matrices with independent entries
• Asymptotic distribution of complex zeros of random analytic functions
• Non-Gaussian Limiting Laws for the Entries of Regular Functions of the Wigner Matrices

Cited by in corpus (1)

• Zeros of random polynomials and its higher derivatives