A Skorohod representation theorem without separability
نویسندگان
چکیده
منابع مشابه
A Skorohod representation theorem without separability
Let (S, d) be a metric space, G a σ-field on S and (μn : n ≥ 0) a sequence of probabilities on G. Suppose G countably generated, the map (x, y) 7→ d(x, y) measurable with respect to G ⊗ G, and μn perfect for n > 0. Say that (μn) has a Skorohod representation if, on some probability space, there are random variables Xn such that Xn ∼ μn for all n ≥ 0 and d(Xn, X0) P −→ 0. It is shown that (μn) h...
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Let (μn : n ≥ 0) be Borel probabilities on a metric space S such that μn → μ0 weakly. Say that Skorohod representation holds if, on some probability space, there are S-valued random variables Xn satisfying Xn ∼ μn for all n and Xn → X0 in probability. By Skorohod’s theorem, Skorohod representation holds (with Xn → X0 almost uniformly) if μ0 is separable. Two results are proved in this paper. Fi...
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Let μn be a probability measure on the Borel σ-field on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]-valued random variables Xn such that Xn ∼ μn for all n ≥ 0 and ‖Xn − X0‖ → 0 in probability, where ‖·‖ is the sup-norm. Such conditions do not require μ0 separable unde...
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Let S be a metric space, G a σ-field of subsets of S and (μn : n ≥ 0) a sequence of probability measures on G. Say that (μn) admits a Skorokhod representation if, on some probability space, there are random variables Xn with values in (S,G) such that Xn ∼ μn for each n ≥ 0 and Xn → X0 in probability. We focus on results of the following type: (μn) has a Skorokhod representation if and only if J...
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With reference to Skorohod representation theorem, it is shown that separability of the limit law cannot be dropped (provided, of course, non separable probabilities exist). An alternative version of the theorem, not requesting separability of the limit, is discussed. A notion of convergence in distribution, extending that of Hoffmann-Jørgensen to non measurable limits, is introduced. For such ...
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ژورنال
عنوان ژورنال: Electronic Communications in Probability
سال: 2013
ISSN: 1083-589X
DOI: 10.1214/ecp.v18-2793