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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)
MSC:
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
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[1] J. Komlos, P. Major, and G. Tusnady, ?An approximation of partial sums of independent RV’s and the sample DF,? Z. Wahrscheinlichkeitstheor. Verw. Geb.,32, No. 2, 111-131 (1975). · Zbl 0308.60029
[2] I. S. Borisov, ?Rate of convergence in invariance principle in linear spaces. Application to empirical measures,? Lect. Notes Math.,1021, 45-58 (1983). · Zbl 0527.60031
[3] I. S. Borisov, ?A new approach to the problem of approximating distributions of sums of independent random variables in linear spaces,? Trudy Mat. Inst., Sib. Sec., Academy of Sciences of the USSR,5, 3-27 (1985).
[4] I. S. Borisov, ?Rate of convergence in the invariance principle for empirical measures,? in: Proc. 1st World Congress Bernoulli Soc., VNU Sci. Press, Amsterdam (1987), pp. 833-836.
[5] P. Massart, ?Rates of convergence in the central limit theorem for empirical processes,? Ann. Inst. Henri Poincaré,22, 381-423 (1986). · Zbl 0615.60032
[6] P. Massart, ?Strong approximation for multivariate empirical and related processes, via KMT constructions,? Ann. Probab.,17, 266-291 (1989). · Zbl 0675.60026
[7] V. I. Kolchinskii (Kol?inski), ?Rates of convergence in the invariance principle for empirical processes,? Festschrift, Yu. Prokhorov et al. (eds.), VSP/Mokslas (1991).
[8] E. Gine and J. Zinn, ?Some limit theorems for empirical processes,? Ann. Probab.,12, 929-998 (1984). · Zbl 0553.60037
[9] V. N. Vapnik and A. Ya. Chervonenkis, ?The uniform convergence of frequencies of the appearance of events to their probabilities,? Teor. Veroyatn. Primen.,16, 264-279 (1971). · Zbl 0247.60005
[10] V. N. Vapnik and A. Ya. Chervonenkis, ?Necessary and sufficient condition of the uniform convergence of empirical means,? Teor. Veroyatn. Primen.,26, 543-563 (1981). · Zbl 0471.60041
[11] V. I. Kolchinskii, ?The central limit theorem for empirical measures,? Teor. Veroyatn. Mat. Stat., Kiev,24, 63-75 (1981).
[12] V. I. Kolchinskii, ?Functional limit theorems and empirical entropy. I,? Teor. Veroyatn. Mat. Stat., Kiev,33, 35-45 (1985).
[13] V. I. Kolchinskii, ?Functional limit thoerems and empirical entropy. II,? Teor. Veroyatn. Mat. Stat., Kiev,34, 81-93 (1986).
[14] L. Le Cam, ?A remark on empirical processes,? in: Festschrift for E. L. Lehmann in Honor of His Sixty-Fifth Birthday, Belmont, California, Wadsworth (1983), pp. 305-327.
[15] E. Gine and J. Zinn, ?Lectures on the central limit theorem for empirical processes,? Lect. Notes Math.,1221, 50-113 (1986).
[16] M. Talagrand, ?Classes de Donsker et ensemble pulverisés,? C. R. Acad. Sci. Paris, Ser. I, 161-163 (1985). · Zbl 0575.60036
[17] R. M. Dudley, ?Central limit theorems for empirical measures,? Ann. Probab.,6, 899-929 (1978). · Zbl 0404.60016
[18] A. N. Zhdanov and E. A. Sevast’yanov, ?Approximative and differential properties of measurable sets,? Mat. Sb.,121, 403-422 (1983).
[19] R. M. Dudley, ?Metric entropy of some classes of sets with differentiable boundaries,? J. Approx. Theory,10, 227-236 (1974). · Zbl 0275.41011
[20] S. Csörgö, ?Limit behaviour of the empirical characteristic function,? Ann. Probab.,9, 130-144 (1981). · Zbl 0453.60025
[21] M. B. Marcus and W. Philipp, ?Almost sure invariance principles for sums of B-valued random variables with applications to random Fourier series and the empirical characteristic process,? Trans. Am. Math. Soc.,269, No. 1, 67-90 (1982). · Zbl 0485.60030
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