Equidistribution of zeros of random polynomials

We study the asymptotic distribution of zeros for the random polynomials Pn(z)=∑k=0nAkBk(z), where {Ak}k=0∞ are non-trivial i.i.d. complex random variables. Polynomials {Bk}k=0∞ are deterministic, and are selected from a standard basis such as Szegő, Bergman, or Faber polynomials associated with a Jordan domain G bounded by an analytic curve. We show that the zero counting measures of Pn converge almost surely to the equilibrium measure on the boundary of G if and only if E[log+|A0|]<∞.