  We show that repulsive random variables can yield Monte Carlo methods with
faster convergence rates than the typical $N^{-1/2}$, where $N$ is the number
of integrand evaluations. More precisely, we propose stochastic numerical
quadratures involving determinantal point processes associated with
multivariate orthogonal polynomials, and we obtain root mean square errors that
decrease as $N^{-(1+1/d)/2}$, where $d$ is the dimension of the ambient space.
First, we prove a central limit theorem (CLT) for the linear statistics of a
class of determinantal point processes, when the reference measure is a product
measure supported on a hypercube, which satisfies the Nevai-class regularity
condition, a result which may be of independent interest. Next, we introduce a
Monte Carlo method based on these determinantal point processes, and prove a
CLT with explicit limiting variance for the quadrature error, when the
reference measure satisfies a stronger regularity condition. As a corollary, by
taking a specific reference measure and using a construction similar to
importance sampling, we obtain a general Monte Carlo method, which applies to
any measure with continuously derivable density. Loosely speaking, our method
can be interpreted as a stochastic counterpart to Gaussian quadrature, which,
at the price of some convergence rate, is easily generalizable to any dimension
and has a more explicit error term.
