Journal ArticleDOI
Almost sure weak convergence of random probability measures
TLDR
In this paper, the authors considered weak weak convergence of random probability measures on a metric space S and showed that for S = T ∞ with T Radon, a.s. convergence of μ n (f) is sufficient for (i) and (ii) implies (iii) while the converse is not true.Abstract:
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.read more
Citations
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Journal ArticleDOI
Convergence of Probability Measures
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Book
Mathematics of Two-Dimensional Turbulence
Sergej B. Kuksin,Armen Shirikyan +1 more
TL;DR: In this article, the uniqueness of stationary measure and mixing is discussed, as well as the limiting theorems of the Navier-Stokes equations and the Inviscid limit.
Journal ArticleDOI
The correlated pseudomarginal method
TL;DR: The correlated pseudomarginal method (CSM) as discussed by the authors is a modification of the pseudo-argininal method using a likelihood ratio estimator computed by using two correlated likelihood estimators.
Journal ArticleDOI
Hybrid Dirichlet mixture models for functional data
TL;DR: In this article, the authors propose a Bayesian mixture model for dimension reduction by representing the sample of n curves through a smaller set of canonical curves, and propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors.
Journal ArticleDOI
Posterior convergence for approximated unknowns in non-Gaussian statistical inverse problems
TL;DR: In this article, the authors studied the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect observation in the Bayesian framework and showed that with the help of the generalized Bayes formula, the question of convergence of posterior distributions is returned to the convergence of the finite-dimensional approximations of the unknown.
References
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Book
Convergence of Probability Measures
TL;DR: Weak Convergence in Metric Spaces as discussed by the authors is one of the most common modes of convergence in metric spaces, and it can be seen as a form of weak convergence in metric space.
Journal ArticleDOI
Convergence of Probability Measures
TL;DR: Convergence of Probability Measures as mentioned in this paper is a well-known convergence of probability measures. But it does not consider the relationship between probability measures and the probability distribution of probabilities.
Journal ArticleDOI
Distributional results for means of normalized random measures with independent increments
TL;DR: In this paper, the authors consider the problem of determining the distribution of means of random probability measures which are obtained by normalizing increasing additive processes and find a solution by resorting to a well-known inversion formula for characteristic functions due to Gurland.
Book
Random Probability Measures on Polish Spaces
TL;DR: This paper presents Prohorov Theory for Random Probability Measures and the Narrow Topology on Non-Random Measures and some Ergodic theory for Random Dynamical Systems, which addresses the role of randomness in the design of deterministic systems.
Journal ArticleDOI
Limit theorems for a class of identically distributed random variables
TL;DR: In this paper, a new type of stochastic dependence for a sequence of random variables is introduced and studied, and it is shown that (Xn)n ≥ 1 is exchangeable if and only if (Xτ(n))n≥1 is c.i.d.