Random polynomials in several complex variables

Abstract

We generalize some previous results on random polynomials in several complex variables. A standard setting is to consider random polynomials \({H_n}(z): = \sum\nolimits_{j = 1}^{{m_n}} {{a_j}{p_j}} \) that are linear combinations of basis polynomials {p_j} with i.i.d. complex random variable coefficients {a_j} where {p_j} form an orthonormal basis for a Bernstein-Markov measure on a compact set \(K \subset {\mathbb{C}^d}\). Here m_n is the dimension of \({{\cal P}_n}\), the holomorphic polynomials of degree at most n in \({\mathbb{C}^d}\). We consider more general bases {p_j}, which include, e.g., higher-dimensional generalizations of Fekete polynomials. Moreover we allow \({H_n}(z): = \sum\nolimits_{j = 1}^{{m_n}} {{a_{nj}}{p_{nj}}(z)} \), i.e., we have an array of basis polynomials {p_j} and random coefficients {a_nj}. This always occurs in a weighted situation. We prove results on convergence in probability and on almost sure convergence of \({1 \over n}\log |{H_n}|\) in \(L_{{\rm{loc}}}^1({\mathbb{C}^d})\) to the (weighted) extremal plurisubharmonic function for K. We aim for weakest possible sufficient conditions on the random coefficients to guarantee convergence.