Skorohod Representation Theorem Via Disintegrations
نویسندگان
چکیده
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. First, Skorohod representation may fail if μ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of μ0 separable or not, Skorohod representation holds if W (μn, μ0)→ 0 whereW is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W ), disintegrable probability measures
منابع مشابه
Two Remarks on Skorohod Representation Theorem
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 ...
متن کاملOpen Mappings of Probability Measures and the Skorohod Representation Theorem
We prove that for a broad class of spaces X and Y (including all Souslin spaces), every open surjective mapping f : X ! Y induces the open mapping 7 ! f ?1 between the spaces of probability measures P(X) and P(Y). Connections with the Skorohod representation theorem and its generalizations are discussed.
متن کاملA 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...
متن کاملIntegral and local limit theorems for level crossings of diffusions and the Skorohod problem
Using a new technique, based on the regularization of a càdlàg process via the double Skorohod map, we obtain limit theorems for integrated numbers of level crossings of diffusions. The results are related to the recent results on the limit theorems for the truncated variation. We also extend to diffusions the classical result of Kasahara on the “local" limit theorem for the number of crossings...
متن کاملA Skorohod Representation Theorem for Uniform Distance
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...
متن کامل