Almost sure weak convergence of random probability measures

By Patrizia Berti, Luca Pratelli, Pietro Rigo. 2006

69 citations recorded by the source corpus.

Given a sequence (μ n ) of random probability measures on a metric space S, consider the conditions: (i) μ n →μ (weakly) a.s. for some random probability measure μ on S; (ii) μ n (f) converges a.s. for all f∈C b (S). Then, (i) implies (ii), while the converse is not true, even if S is separable. For (i) and (ii) to be equivalent, it is enough that S is Radon (i.e. each probability on the Borel sets of S is tight) or that the sequence (P μ n ) is tight, where Pμ n (·)=E(μ n (·)). In particular, (i)⇔(ii) in case S is Polish. The latter result is still available if a.s. convergence is weakened into convergence in probability. In case S=T ∞ with T Radon, a.s. convergence of μ n (f), for those f∈C b (S) which are finite products of elements of C b (T), is sufficient for (i). In case and the limit μ is given in advance, a.s. convergence of characteristic functions is enough for μ n →μ (weakly) a.s. Almost sure weak convergence of random probability measures.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7edae071a297ac3a29ec28b0775762ed807b4d7d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Hybrid Dirichlet mixture models for functional data (2009, background): “Finding general conditions on the qk parameters such that Qk converges to a probability measure is beyond the scope of this paper (see corollary 2.5 in Berti et al. (2006) ).”
The correlated pseudomarginal method (2015, background): “Here πT .θ/ is interpreted as the density of a YT -measurable random probability measure; see, for example, Berti et al. (2006) and Crauel (2003) for a formal definition.”