Let $(Z^{ (n)}_k)_{ 1 \leq k \leq n}$ be a random set of points and let $\mu_n$ be its \emph{ empirical measure}: $$\mu_n = \frac{ 1}{ n} \sum_{ k=1}^n \delta_{ Z^{ (n)}_k}. $$ Let $$P_n(z) := (z - Z^{ (n)}_1)\cdots (z - Z^{ (n)}_n)\quad \text{ and}\quad Q_n (z) := \sum_{ k=1}^n \gamma^{ (n)}_k \prod_{ 1 \leq j \leq n, j \neq k} (z- Z^{ (n)}_j), $$ where $(\gamma^{ (n)}_k)_{ 1 \leq k \leq n}$ are independent, i.i.d. random variables with Gamma distribution of parameter $\beta/2$, for some fixed $\beta > 0$. We prove that in the case where $\mu_n$ almost surely tends to $\mu$ when $n \rightarrow \infty$, the empirical measure of the complex zeros of the \emph{ randomized derivative} $Q_n$ also converges almost surely to $\mu$ when $n$ tends to infinity. Furthermore, for $k = o(n / \log n)$, we obtain that the zeros of the $k-$th \emph{ randomized derivative} of $P_n$ converge to the limiting measure $\mu$ in the same sense. We also derive the same conclusion for a variant of the randomized derivative related to the unit circle.
Categories: math.PR