Skorohod Representation Theorem Via Disintegrations

Patrizia Berti, Luca Pratelli and Pietro Rigo
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Patrizia Berti: Università di Modena e Reggio Emilia
Luca Pratelli: Accademia Navale di Livorno
Pietro Rigo: Department of Economics and Quantitative Methods, University of Pavia

No 104, Quaderni di Dipartimento from University of Pavia, Department of Economics and Quantitative Methods

Abstract: 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 where W 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 are fundamental.

Keywords: Disintegration; Separable probability measure; Skorohod representation theorem; Wasserstein distance; Weak convergence of probability measures. (search for similar items in EconPapers)
Pages: 10 pages
Date: 2009-10
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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