NDLI: Skorohod representation on a given probability space

Content Provider	Springer Nature Link
Author	Berti, Patrizia Rigo, Pietro Pratelli, Luca
Copyright Year	2006
Abstract	Let $$(\Omega,\mathcal{A},P)$$ be a probability space, S a metric space, μ a probability measure on the Borel σ-field of S, and $$X_n:\Omega\rightarrow S$$ an arbitrary map, n = 1,2,.... If μ is tight and X n converges in distribution to μ (in Hoffmann–Jørgensen’s sense), then X∼μ for some S-valued random variable X on $$(\Omega,\mathcal{A},P)$$ . If, in addition, the X n are measurable and tight, there are S-valued random variables $$\overset{\sim}{X}_n$$ and X, defined on $$(\Omega,\mathcal{A},P)$$ , such that $$\overset{\sim}{X}_n\sim X_n$$ , X∼μ, and $$\overset{\sim}{X}_{n_k}\rightarrow X$$ a.s. for some subsequence (n k ). Further, $$\overset{\sim}{X}_n\rightarrow X$$ a.s. (without need of taking subsequences) if μ{x} = 0 for all x, or if P(X n = x) = 0 for some n and all x. When P is perfect, the tightness assumption can be weakened into separability up to extending P to $$\sigma(\mathcal{A}\cup\{H\})$$ for some H⊂Ω with P *(H) = 1. As a consequence, in applying Skorohod representation theorem with separable probability measures, the Skorohod space can be taken $$((0,1),\sigma(\mathcal{U}\cup\{H\}),m_H)$$ , for some H⊂ (0,1) with outer Lebesgue measure 1, where $$\mathcal{U}$$ is the Borel σ-field on (0,1) and m H the only extension of Lebesgue measure such that m H (H) = 1. In order to prove the previous results, it is also shown that, if X n converges in distribution to a separable limit, then X n k converges stably for some subsequence (n k ).
Ending Page	288
Page Count	12
Starting Page	277
File Format	PDF
ISSN	01788051
e-ISSN	14322064
Journal	Probability Theory and Related Fields
Issue Number	3
Volume Number	137
Language	English
Publisher	Springer-Verlag
Publisher Date	2006-07-26
Publisher Place	Berlin/Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Weak convergence of probability measures Empirical process Mathematical and Computational Physics Convergence of probability measures Non measurable random element Probability Theory and Stochastic Processes Probabilistic measure theory Quantitative Finance Statistics for Business/Economics/Mathematical Finance/Insurance Operations Research/Decision Theory Skorohod representation theorem Stable convergence Mathematical Biology in General Axioms; other general questions
Content Type	Text
Resource Type	Article
Subject	Statistics and Probability Analysis Statistics, Probability and Uncertainty

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