We study the asymptotic distribution of zeros for the random polynomials $P_n(z) = \sum_{ k=0}^n A_k B_k(z)$, where $\{ A_k\}_{ k=0}^{ \infty}$ are non-trivial i.i.d. complex random variables. Polynomials $\{ B_k\}_{ k=0}^{ \infty}$ are deterministic, and are selected from a standard basis such as Szeg\H{ o}, Bergman, or Faber polynomials associated with a Jordan domain $G$ bounded by an analytic curve. We show that the zero counting measures of $P_n$ converge almost surely to the equilibrium measure on the boundary of $G$ if and only if $\mathbb{ E}[\log^+|A_0|]<\infty$.
Categories: math.CV math.PR