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Accuracy of the approximation of an empirical process by a Brownian bridge. (English. Russian original) Zbl 0786.60040
Sib. Math. J. 32, No. 4, 578-588 (1991); translation from Sib. Mat. Zh. 32, No. 4(188), 48-60 (1991).
Let \((X,{\mathcal A},P)\) be a probability space. Let \(P_ n\) be empirical measures for \(P\). We ask that \(Z_ n\) is an empirical process if \(Z_ n=n^{1/2}(P_ n-P)\). In a number of papers, the Kolmós-Major- Tusnády theorem [J. Kolmós, P. Major and G. Tusnády, Z. Wahrscheinlichkeitstheorie Verw. Geb. 32, 111-131 (1975; Zbl 0308.60029)] is generalized to the case of an empirical process on measurable spaces and functions [see for example I. S. Borisov, Probability theory and mathematical statistics, Proc. 4th USSR-Jap. Symp., Tbilisi/USSR 1982, Lect. Notes Math. 1021, 45-58 (1983; Zbl 0527.60031) or P. Massart, Ann. Probab. 17, No. 1, 266-291 (1989; Zbl 0675.60026)]. In this paper, the author proves some results already announced in the paper reviewed above. These results are a continuation of a Borisov-Massart approach.
Reviewer: D.Aissani (Bejaia)
60F17 Functional limit theorems; invariance principles
60J65 Brownian motion
60B12 Limit theorems for vector-valued random variables (infinite-dimensional case)
60F05 Central limit and other weak theorems
Full Text: DOI
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