References
A. Yu. Zaîtsev, “Estimation of the Lévy-Prokhorov distance in the central limit theorem for random vectors with finite exponential moments,” Teor. Veroyatnost. i Primenen.,31, No. 2, 203–220 (1986).
A. A. Borovkov, “On the rate of convergence in the invariance principle,” Teor. Veroyatnost. i Primenen.,18, No. 2, 217–234 (1973).
M. Csőrgő and P. Révész, Strong Approximations in Probability and Statistics, Academic Press, New York (1981).
S. Csőrgő and P. Hall, “The Komlós-Major-Tusnády approximations and their applications,” Austral. J. Statist.,26, No. 2, 189–218 (1984).
Yu. V. Prokhorov, “Convergence of random processes and the limit theorems of probability theory,” Teor. Veroyatnost. i Primenen.,1, No. 2, 157–214 (1956).
A. V. Skorokhod, Studies in the Theory of Random Processes [in Russian], Kievsk. Univ., Kiev (1961).
J. Komlós, P. Major, and G. Tusnády, “An approximation of partial sums of independent RV's and the sample DF. I,” Z. Wahrscheinlichkeitstheor. Verw. Geb., No. 32, 111–131 (1975).
J. Komlós, P. Major, and G. Tusnády, “An approximation of partial sums of independent RV's and the sample DF. II,” Z. Wahrscheinlichkeitstheor. Verw. Geb., No. 34, 34–58 (1976).
A. I. Sakhanenko, “On the rate of convergence in the invariance principle for nonidentically distributed variables with exponential moments,” in: Trudy Inst. Mat. (Novosibirsk), Novosibirsk,3, 4–49 (1984).
E. Berger, Fast Sichere Approximation von Partialsummen Unabhängiger und Stationärer Ergodischer Folgen von Zufallsvectoren, Dissertation, Universität Göttingen (1982).
I. Berkes and W. Philipp, “Approximation theorems for independent and weakly dependent random vectors,” Ann. Probab.,7, 29–54 (1979).
U. Einmahl, “A useful estimate in the multidimensional invariance principle,” Probab. Theory Related Fields,76, No. 1, 81–101 (1987).
U. Einmahl, “Strong invariance principles for partial sums of independent random vectors,” Ann. Probab.,15, 1419–1440 (1987).
W. Philipp, “Almost sure invariance principles for sums of B-valued random variables,” Lecture Notes in Math.,709, 171–193 (1979).
U. Einmahl, “Extensions of results of Komlós, Major and Tusnády to the multivariate case,” J. Multivariate Anal.,28, 20–68 (1989).
A. Yu. Zaîtsev, “A multidimensional version of the Hungurian construction,” in: Abstracts: Second All-Russian School-Colloquium on Stochastic Methods (Îoshkar-Ola, 1995), TVP, Moscow, 1995, pp. 54–55.
A. Yu. Zaîtsev, Multidimensional Version of the Results of Komlós, Major and Tusnády for Vectors with Finite Exponential Moments [Preprint], Univ. Bielefeld 95-055, FRG, Bielefeld (1995).
V. V. Yurinsky, “On approximation of convolutions by normal laws,” Teor. Veroyatnost. i Primenen.,22, No. 4, 653–667 (1977).
A. Yu. Zaîtsev, “On the Gaussian approximation of convolutions under multidimensional analogues of S. N. Bernstein inequality conditions,” Probab. Theory Related Fields,74, No. 4, 535–566 (1987).
V. A. Statulevičius, “On large deviations,” Z. Wahrscheinlichkeitstheor. Verw. Geb., No. 6, 133–144 (1966).
L. Saulis and V. Statulyavichus, Limit Theorem on Large Deviations [in Russian], Mokslas, Vil'nyus (1991).
A. Yu. Zaîtsev, “An improvement of the U. Einmahl estimate in the multidimensional invariance principle,” in: I. A. Ibragimov and A. Yu. Zaîtsev (eds.), Probability Theory and Mathematical Statistics, Proceedings of the Euler Institute Seminars Dedicated to the Memory of Kolmogorov, Gordon and Breach, Amsterdam, 1996, pp. 109–116.
Author information
Authors and Affiliations
Additional information
The research is financially supported by the Alexander von Humbolt Foundation, the Russian Foundation for Basic Research (Grant 93-11-01454), the INTAS (Grant 93-1585), and the International Science Foundation (Grant R3600).
Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 37, No. 4, pp. 807–831, July–August, 1996.
Rights and permissions
About this article
Cite this article
Zaîtsev, A.Y. Estimates for the quantiles of smooth conditional distributions and the multidimensional invariance principle. Sib Math J 37, 706–729 (1996). https://doi.org/10.1007/BF02104663
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02104663