×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[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 · doi:10.1007/BF00533093
[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 · doi:10.1007/BFb0072902
[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 · doi:10.1214/aop/1176991508
[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 · doi:10.1214/aop/1176993138
[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). · doi:10.1007/BFb0099111
[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 · doi:10.1214/aop/1176995384
[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 · doi:10.1016/0021-9045(74)90120-8
[20] S. Csörgö, ?Limit behaviour of the empirical characteristic function,? Ann. Probab.,9, 130-144 (1981). · Zbl 0453.60025 · doi:10.1214/aop/1176994513
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.