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On normal approximation of large products of functions: A refinement of Blackwell’s result. (English) Zbl 1005.62013
From the paper: D. Blackwell [Approximate normality of large products. Tech. Rep. No. 54, Dpt. Stat., Univ. California, Berkeley (1995)], using a simple elementary way, proved that under some conditions standardized products of large numbers of smooth positive functions are close to $$\exp(-x^2/2)$$. Moreover, he gave an estimation of this closeness. In the present note we observe that, under additional smoothness conditions, a little more effort provides an asymptotic expansion making this closeness still more precise. This result is closely related to the Bernstein-von Mises theorem on the normal approximation of posterior distributions.
##### MSC:
 62E20 Asymptotic distribution theory in statistics 62F15 Bayesian inference 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)
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