Critical points of random polynomials with independent identically distributed roots

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Let X₁,X₂,... be independent identically distributed random variables with values in. Denote by μ the probability distribution of X₁. Consider a random polynomial Pₙ(z)=(z-X₁)...(z-Xₙ). We prove a conjecture of Pemantle and Rivin [arXiv:1109.5975] that the empirical measure μₙ:= 1n-1∑ₚₙ'₍z₎₌₀ δz counting the complex zeros of the derivative Pₙ' converges in probability to μ, as n→∞.