  Randomized neural networks (NNs) are an interesting alternative to
conventional NNs that are more used for data modeling. The random vector
functional-link (RVFL) network is an established and theoretically
well-grounded randomized learning model. A key theoretical result for RVFL
networks is that they provide universal approximation for continuous maps, on
average, almost surely. We specialize and modify this result, and show that
RFVL networks can provide functional approximations that converge in
Kullback-Leibler divergence, when the target function is a probability density
function. Expanding on the approximation results, we demonstrate the the RFVL
networks lead to a simple randomized mixture model (MM) construction for
density estimation from random data. An expectation-maximization (EM) algorithm
is derived for the maximum likelihood estimation of our randomized MM. The EM
algorithm is proved to be globally convergent and the maximum likelihood
estimator is proved to be consistent. A set of simulation studies is given to
provide empirical evidence towards our approximation and density estimation
results.
