Random Polynomials in Several Complex Variables

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials Hn(z) := ∑mn j=1 ajpj(z) that are linear combinations of basis polynomials {pj} with i.i.d. complex random variable coefficients {aj} where {pj} form an orthonormal basis for a Bernstein-Markov measure on a compact set K ⊂ Cd. Here mn is the dimension of Pn, the holomorphic polynomials of degree at most n in Cd. We consider more general bases {pj}, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow Hn(z) := ∑mn j=1 anjpnj(z); i.e., we have an array of basis polynomials {pnj} and random coefficients {anj}. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of 1 n log |Hn| in Lloc(C) to the (weighted) extremal plurisubharmonic function for K. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.

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