A necessary and sufficient condition for convergence of the zeros of random polynomials

Consider random polynomials of the form $G_n = \sum_{i=0}^n \xi_i p_i$, where $\xi_i$ are i.i.d.\ non-degenerate complex variables, and $\{p_i\}$ is a sequence orthonormal with respect to regular measure $\tau$ supported on compact set $K$. We show that zero $G_n$ converges weakly almost surely equilibrium $K$ if only $\mathbb{E} \log(1 + |\xi_0|) < \infty$. This generalizes corresponding result Ibragimov Zaporozhets in case when $p_i(z) z^i$. also probability $\mathbb{P} (|\xi_0| > e^n) o(n^{-1})$. Our proofs rely results from small ball exploit structure general orthogonal polynomials. methods work for sequences asymptotically minimal $L^p(\tau)$, $p \in (0, \infty]$. In particular, $L^p$-minimal (normalized) Faber Fekete fall into this class.