Convergence in probability of all predictive distributions of a sequence is characterized by stable convergence, setwise convergence of marginals, and a second-moment matching condition.
(2012) Limit theorems for empirical processes based on dependent data, Electr
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Weak convergence of predictive distributions
Convergence in probability of all predictive distributions of a sequence is characterized by stable convergence, setwise convergence of marginals, and a second-moment matching condition.