  We present an efficient numerical scheme based on Monte Carlo integration to
approximate statistical solutions of the incompressible Euler equations. The
scheme is based on finite volume methods, which provide a more flexible
framework than previously existing spectral methods for the computation of
statistical solutions for incompressible flows. This finite volume scheme is
rigorously proven to, under experimentally verifiable assumptions, converge in
an appropriate topology and with increasing resolution to a statistical
solution. The convergence obtained is stronger than that of measure-valued
solutions, as it implies convergence of multi-point correlation marginals. We
present results of numerical experiments which support the claim that the
aforementioned assumptions are very natural, and appear to hold in practice.
