Abstract
We establish that Laplace transforms of the posterior Dirichlet process converge to those of the limiting Brownian bridge process in a neighbourhood about zero, uniformly over Glivenko-Cantelli function classes. For real-valued random variables and functions of bounded variation, we strengthen this result to hold for all real numbers. This last result is proved via an explicit strong approximation coupling inequality.
Funding Statement
The research leading to these result was (partly) financed by the NWO Spinoza prize awarded to A.W. van der Vaart by the Netherlands Organisation for Scientific Research (NWO). The research leading to these results has received funding from the European Research Council under ERC Grant Agreement 320637.
Acknowledgments
We would like to thank two referees for their helpful comments, in particular one referee for pointing out a missing step in the proof.
Citation
Kolyan Ray. Aad van der Vaart. "On the Bernstein-von Mises theorem for the Dirichlet process." Electron. J. Statist. 15 (1) 2224 - 2246, 2021. https://doi.org/10.1214/21-EJS1821
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