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Lower bounds on the approximation of the multivariate empirical process. (English) Zbl 0554.60037
It is well-known that if C is the class of rectangles \(0\leq x_ 1\leq a_ 1\), \(0\leq x_ 2\leq a_ 2\) or the class of circular discs then the normalized empirical measure on C behaves like a Brownian bridge. Our main result shows that for these two classes the distances between the normalized empirical measure and the nearest Brownian measure have entirely different order of magnitudes.

60F15 Strong limit theorems
60J65 Brownian motion
Full Text: DOI
[1] Bártfai, P.: Über die Entfernung der Irrfahrtswege. Stud. Sci. Math. Hungar.5, 41–49 (1970) · Zbl 0274.60048
[2] Beck, J.: On a problem of K.F. Roth concerning irregularities of point distribution. Invent. Math.74, 477–487 (1983) · Zbl 0528.10037 · doi:10.1007/BF01394247
[3] Csörgo, M., Révész, P.: A new method to prove Strassen type laws of invariance principle. II. Z. Wahrscheinlichkeitstheor. Verw. Geb.31, 261–269 (1975) · Zbl 0283.60024 · doi:10.1007/BF00532866
[4] Dudley, R.M.: Central limit theorems for empirical measures. Ann. Probab.6, 899–929 (1978); Correction ibid.7, 909–911 (1979) · Zbl 0404.60016 · doi:10.1214/aop/1176995384
[5] Komlós, J., Major, P., Tusnády, G.: An approximation of partial sums of independent r.v.’s and the sample d.f.. Z. Wahrscheinlichkeitstheor. Verw. Geb.32, 111–131 (1975) · Zbl 0308.60029 · doi:10.1007/BF00533093
[6] Olver, F.W.J.: Asymptotics and Special Functions. New York and London: Academic Press 1974 · Zbl 0303.41035
[7] Révész, P.: On strong approximation of the multidimensional empirical process. Ann. Probab.4, 729–743 (1976) · Zbl 0344.60022 · doi:10.1214/aop/1176995981
[8] Révész, P.: Three theorem of multivariate empirical process. Empirical Distributions and Processes. Lect. Notes Math.566, 106–126. Berlin-Heidelberg-New York: Springer 1976
[9] Roth, K.F.: On irregularities of distribution. Mathematika1, 73–79 (1954) · Zbl 0057.28604 · doi:10.1112/S0025579300000541
[10] Schmidt, W.M.: Irregularities of distribution. IV. Invent. Math.7, 55–82 (1969) · Zbl 0172.06402 · doi:10.1007/BF01418774
[11] Tusnády, G.: A remark on the approximation of the sample DF in the multidimensional case. Period. Math. Hung.8, 53–55 (1977) · Zbl 0386.60006 · doi:10.1007/BF02018047
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