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Convergence of conditional expectations given the random variables of a Skorohod representation. (English) Zbl 0957.60003

It is shown that under some technical condition on probability distributions \(P_n\Rightarrow P\) the Skorokhod representations \(X_n, X\) have the following interesting property: For any \(L^2\) random variable \(Z\) on \([0,1]\) the conditional expectations \(E(Z\mid {\mathcal F}^{X_n})\) converge in \(L^2\) to \(E(Z\mid {\mathcal F}^X)\). The proof is carried out by a detailed analysis of Billingsley’s original proof of the Skorokhod representation together with martingale convergence arguments and estimates of conditional expectations in terms of sub-sigma-algebras. In an example it is shown that some additional condition is necessary for the main theorem to hold.

MSC:

60B10 Convergence of probability measures
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References:

[1] Billingsley, P., Probability and Measure (1986), Wiley: Wiley New York · Zbl 0649.60001
[2] Billingsley, P., Weak convergence of measures, CBMS-NSF, (Regional Conf. Ser. in Appl. Math. (1971)), 5
[3] Goggin, E., Convergence in distribution of conditional expectations, Ann. Probab., 22, 1097-1114 (1994) · Zbl 0805.60017
[4] Rogge, L., Uniform inequalities for conditional expectations, Ann. Probab., 2, 486-489 (1974) · Zbl 0285.28010
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