The speed of rational learning

A central result in the rational learning literature is that if the true me asure is absolutely continuous with respect to the beliefs then, given enou gh data, the updated beliefs merge with the true distribution. In this pape r, we show that, under absolute continuity, weak merging occurs fast (at th e rate 1/root t) with density one. Moreover, if weak merging occurs fast en ough (at the rate 1/root t) then absolute continuity holds. These rates are sharp. We also show that, under some conditions, if weak merging occurs at the rate 1/root t then absolute continuity holds.