Variance of set-indexed sums of mixing random variables and weak convergence of set-indexed processes
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. \textit{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 \textit{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.
weak-convergence of continuous-path processes
scientific article; zbMATH DE number 3975998