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Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes. (English) Zbl 0604.60032

Continuing their work ”Characterization of set-indexed Brownian motion and associated conditions for finite-dimensional convergence.” ibid. 14, 802-816 (1986), the authors combine the results obtained there with a tightness lemma to give weak-convergence of continuous-path processes. To achieve this the index set is restricted, requiring that it satisfies a metric entropy condition [cf. R. M. Dudley, ibid. 1, 66-103 (1973; Zbl 0261.60033)] in order that the processes dealt with and their Wiener limit can live in a space of continuous functions.
The tightness proof requires hypotheses on the metric entropy of the index set, and \((2+\epsilon)\)-moments and a (severe) mixing rate for the summands. An extension to the dependent case of a method of R. F. Bass [Z. Wahrscheinlichkeitstheor. Verw. Geb. 70, 591-608 (1985; Zbl 0575.60034)] is essentially used. As results of independent interest we note variance bounds on arbitrary sums of \(Z^ d\)-indexed mixing random variables, and uniform integrability conditions for such sums.
Reviewer: M.Iosifescu

MSC:

60F17 Functional limit theorems; invariance principles
60E15 Inequalities; stochastic orderings
60B10 Convergence of probability measures
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