On weak convergence in $\mathcal{M}\left({H}\right)$: relaxation of a sufficient condition
01 Feb 2014-Vol. 76, Iss: 1, pp 146-149
TL;DR: Theorem 2.2 of Majumdar (Sankhyā 67:670-673, 2005) obtained two conditions that are necessary and jointly sufficient for weak convergence in this paper.
Abstract: Theorem 2.2 of Majumdar (Sankhyā 67:670–673, 2005) obtained two conditions that are necessary and jointly sufficient for weak convergence in \( {\mathcal{M}}\left({H}\right)\). In this note we significantly relax the second condition while maintaining the joint sufficiency.
Citations
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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.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
References
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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.
Abstract: Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.
5,689 citations
Book•
01 Jan 1967
TL;DR: The Borel subsets of a metric space Probability measures in the metric space and probability measures in a metric group Probability measure in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability probabilistic probability measures on $C[0, 1]$ and $D[0-1]$ Bibliographical notes Bibliography List of symbols Author index Subject index as mentioned in this paper
Abstract: The Borel subsets of a metric space Probability measures in a metric space Probability measures in a metric group Probability measures in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability Probability measures in a Hilbert space Probability measures on $C[0,1]$ and $D[0,1]$ Bibliographical notes Bibliography List of symbols Author index Subject index.
2,667 citations
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.
Abstract: Weak Convergence in Metric Spaces. The Space C. The Space D. Dependent Variables. Other Modes of Convergence. Appendix. Some Notes on the Problems. Bibliographical Notes. Bibliography. Index.
1,564 citations
01 Jan 1967
TL;DR: In this article, the authors provide an overview on probability measures in a metric space and present a smaller class of measures on metric spaces called tight measures, which have the property that they are determined by their values for compact sets.
Abstract: This chapter provides an overview on probability measures in a metric space. A measure μ on a metric space means a countably additive nonnegative set function μ on the class of Borel sets B x with the property that μ ( X ) = 1 . In a metric space, a measure μ is uniquely determined by its values for the topologically important sets such as closed sets or open sets. The chapter also presents a smaller class of measures on metric spaces called the tight measures. Tight measures have the property that they are determined by their values for compact sets. A measure μ on a metric space X is said to be tight if for each ɛ > 0, there exists a compact set K 2 ⊊ X such that μ ( X − κ ɛ ) ɛ . A very extensive class of metric spaces has the property that every measure is tight.
759 citations