A. Y. Lo, V. V. Sazonov

On Normal Approximation of Large Products of Functions: a Refinement of Blackwell's Result

abstract:
An asymptotic expansion for the approximation of standardized products of large numbers of smooth positive functions by $\exp (- x^2/2)$ is given. This result is closely related to the Bernstein--von Mises theorem on the normal approximation of posterior distributions.