# Weak convergence and the exponential rate of concentration for posterior density functions

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## Abstract

Consider a Bayesian analysis of a parameter vector, [theta], based on n i.i.d. multivariate measurements. We establish weak convergence of a sequence of parameter values that arises in applying the mean value theorem for integrals to the marginal densities of the sequence of observed vectors. We apply this weak convergence theorem to obtain a finite-sample result characterizing the rate of change of the shape of the posterior density as the number of observations increases.

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