×

zbMATH — the first resource for mathematics

Estimates for the quantiles of smooth conditional distributions and the multidimensional invariance principle. (English. Russian original) Zbl 0881.60034
Sib. Math. J. 37, No. 4, 706-729 (1996); translation from Sib. Mat. Zh. 37, No. 4, 807-831 (1996).
This is the first half of the proof of the theorem given by the author in 1995 in his Bielefeld University preprint devoted to the rates in invariance principles for independent centered random vectors (r.v.’s) \(\xi_1,\ldots,\xi_N\in\mathbb{R}^d\). Their laws \({\mathcal L}(\xi_k)\) belong to a certain class \({\mathcal A}_d(\tau)\) with \(\tau\geq 1\) and \(\text{cov} \xi_k=I\), \(k=1,\ldots,n\). The claim is that for \(\alpha>0\) one can construct on a probability space \((\Omega,{\mathcal F},\mathbb{P})\) independent r.v.’s \(X_1,\ldots,X_n\) and independent standard Gaussian r.v.’s \(Y_1,\ldots,Y_n\), s.t. \({\mathcal L}(X_k)={\mathcal L}(\xi_k)\), \(k=1,\ldots,n\), and \[ E\exp\Biggl(\frac{c_1(\alpha)\Delta(X,Y)}{\tau d^3\log^+d}\Biggr)\leq\exp\Bigl(c_2(\alpha)d^{9/4+\alpha}(\log^+(n/\tau^2))\Bigr) \] where \(c_1(\alpha),c_2(\alpha)>0\) depend only on \(\alpha\) and \(\Delta(X,Y)=\max_{1\leq k\leq n}\Bigl|\sum_{j=1}^k X_j- \sum_{j=1}^k Y_j\Bigr|\). This theorem is a refinement of a result by U. Einmahl [J. Multivariate Anal. 28, No. 1, 20-68 (1989; Zbl 0676.60038)]. The proof is based on the study of quantiles of smooth conditional distributions for vectors whose laws belong to \({\mathcal A}_d(\tau)\).

MSC:
60F17 Functional limit theorems; invariance principles
62A01 Foundations and philosophical topics in statistics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] 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).
[2] A. A. Borovkov, ”On the rate of convergence in the invariance principle,” Teor. Veroyatnost. i Primenen.,18, No. 2, 217–234 (1973). · Zbl 0323.60031
[3] M. Csorgo and P. Révész, Strong Approximations in Probability and Statistics, Academic Press, New York (1981).
[4] S. Csorgo and P. Hall, ”The Komlós-Major-Tusnády approximations and their applications,” Austral. J. Statist.,26, No. 2, 189–218 (1984). · Zbl 0557.60028 · doi:10.1111/j.1467-842X.1984.tb01233.x
[5] Yu. V. Prokhorov, ”Convergence of random processes and the limit theorems of probability theory,” Teor. Veroyatnost. i Primenen.,1, No. 2, 157–214 (1956).
[6] A. V. Skorokhod, Studies in the Theory of Random Processes [in Russian], Kievsk. Univ., Kiev (1961). · Zbl 0215.53501
[7] 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). · Zbl 0308.60029 · doi:10.1007/BF00533093
[8] 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). · Zbl 0307.60045
[9] 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). · Zbl 0541.60024
[10] E. Berger, Fast Sichere Approximation von Partialsummen Unabhängiger und Stationärer Ergodischer Folgen von Zufallsvectoren, Dissertation, Universität Göttingen (1982).
[11] I. Berkes and W. Philipp, ”Approximation theorems for independent and weakly dependent random vectors,” Ann. Probab.,7, 29–54 (1979). · Zbl 0392.60024 · doi:10.1214/aop/1176995146
[12] U. Einmahl, ”A useful estimate in the multidimensional invariance principle,” Probab. Theory Related Fields,76, No. 1, 81–101 (1987). · Zbl 0608.60029 · doi:10.1007/BF00390277
[13] U. Einmahl, ”Strong invariance principles for partial sums of independent random vectors,” Ann. Probab.,15, 1419–1440 (1987). · Zbl 0637.60041 · doi:10.1214/aop/1176991985
[14] W. Philipp, ”Almost sure invariance principles for sums of B-valued random variables,” Lecture Notes in Math.,709, 171–193 (1979). · Zbl 0418.60013 · doi:10.1007/BFb0071957
[15] U. Einmahl, ”Extensions of results of Komlós, Major and Tusnády to the multivariate case,” J. Multivariate Anal.,28, 20–68 (1989). · Zbl 0676.60038 · doi:10.1016/0047-259X(89)90097-3
[16] 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.
[17] 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).
[18] V. V. Yurinsky, ”On approximation of convolutions by normal laws,” Teor. Veroyatnost. i Primenen.,22, No. 4, 653–667 (1977). · Zbl 0399.60022
[19] 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). · Zbl 0612.60031 · doi:10.1007/BF00363515
[20] V. A. Statulevičius, ”On large deviations,” Z. Wahrscheinlichkeitstheor. Verw. Geb., No. 6, 133–144 (1966). · Zbl 0158.36207
[21] L. Saulis and V. Statulyavichus, Limit Theorem on Large Deviations [in Russian], Mokslas, Vil’nyus (1991).
[22] 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. · Zbl 0873.60020
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.