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A weak convergence theorem for continuum structure functions. (English) Zbl 0707.60005

Authors’ abstract: A continuum structure function (CSF) is a nondecreasing mapping from the unit hypercube to the unit interval. It is shown that if X(t)\(\to^{{\mathcal D}}X\), an absolutely continuous random vector, as \(t\to \infty\), and if \(\{\gamma_ t\}\) is a class of CSF’s such that \(\gamma_ t\to \gamma\) either (i) pointwise or (ii) in a quasi-Skorokhod topology, then \(\gamma_ t(X(t))\to^{{\mathcal D}}\gamma (X)\) as \(t\to \infty\).
Reviewer: A.J.Rachkauskas

MSC:

60B10 Convergence of probability measures
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
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References:

[1] Baxter, L. A., Continuum structures, I, J. Appl. Probab., 21, 802-815 (1984) · Zbl 0552.60081
[2] Baxter, L. A., Continuum structures, II, (Math. Proc. Cambridge Philos. Soc., 99 (1986)), 331-338 · Zbl 0594.60088
[3] Billingsley, P., Convergence of Probability Measures (1968), Wiley: Wiley New York · Zbl 0172.21201
[4] Block, H. W.; Savits, T. H., Continuous multistate structure functions, Oper. Res., 32, 703-714 (1984) · Zbl 0544.90041
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