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

Let $ p_n(x) $ be a random polynomial of degree $n$ and $\{Z^{(n)}_j\}_{j=1}^n$ and $\{X^{n, k}_j\}_{j=1}^{n-k}, k<n$, 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 $\frac{1}{a_n} \left[ f\left( \frac {Z^{(n)}_1}{b_n} \right)+ \cdots + f \left(\frac {Z^{(n)}_n}{b_n} \right) \right]$ associated with $\{Z^{(n)}_j\}$ has a limit as $n\to\infty$ at some mode of convergence, the linear statistics associated with $\{X^{n, k}_j\}$ 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 $\{Z^{(n)}_j\}$ and the sequences $\{a_n\}$ and $\{b_n\}$ of positive numbers satisfy some mild conditions.