  We generalize some previous results on random polynomials in several complex
variables. A standard setting is to consider random polynomials
$H_n(z):=\sum_{j=1}^{m_n} a_jp_j(z)$ 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 $K\subset {\bf C}^d$. Here $m_n$ is the dimension of
$\mathcal P_n$, the holomorphic polynomials of degree at most $n$ in ${\bf
C}^d$. We consider more general bases $\{p_j\}$, which include, e.g.,
higher-dimensional generalizations of Fekete polynomials. Moreover we allow
$H_n(z):=\sum_{j=1}^{m_n} a_{nj}p_{nj}(z)$; i.e., we have an array of basis
polynomials $\{p_{nj}\}$ 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 $\frac{1}{n}\log |H_n|$ in $L^1_{loc}({\bf
C}^d)$ to the (weighted) extremal plurisubharmonic function for $K$. We aim for
weakest possible sufficient conditions on the random coefficients to guarantee
convergence.
