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This article is cited in 9 scientific papers (total in 9 papers)
Bounds for the Rate of Strong Approximation in the Multidimensional Invariance Principle
F. Götzea, A. Yu. Zaitsevb
a Bielefeld University, Department of Mathematics
b St. Petersburg Department of V. A. Steklov Institute of Mathematics, Russian Academy of Sciences
Abstract:
The goal of this paper is to derive consequences of the result of Zaitsev [Theory Probab. Appl., 45 (2001), pp. 624–642; 46 (2002), pp. 490–514; 676–698]. We establish bounds for the rate of strong Gaussian approximation of sums of independent $\mathbf{R}^d$-valued random vectors $\xi_j$ having finite moments $\mathbf{E}\|\xi_j\|^\gamma$, $\gamma\ge 2$. A multidimensional version of the results of Sakhanenko [Trudy Inst. Mat., 5 (1985), pp. 27–44 (in Russian)] is obtained.
Keywords:
multidimensional invariance principle, strong approximation, sums of independent random vectors.
DOI:
https://doi.org/10.4213/tvp2484
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English version:
Theory of Probability and its Applications, 2009, 53:1, 59–80
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Received: 31.07.2007
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Citation:
F. Götze, A. Yu. Zaitsev, “Bounds for the Rate of Strong Approximation in the Multidimensional Invariance Principle”, Teor. Veroyatnost. i Primenen., 53:1 (2008), 100–123; Theory Probab. Appl., 53:1 (2009), 59–80
Citation in format AMSBIB
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A. Yu. Zaitsev, “The rate of Gaussian strong approximation for the sums of i.i.d. multidimensional random vectors”, J. Math. Sci. (N. Y.), 163:4 (2010), 399–408
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F. Götze, A. Yu. Zaitsev, “Rates of approximation in the multidimensional invariance principle for sums of i.i.d. random vectors with finite moments”, J. Math. Sci. (N. Y.), 167:4 (2010), 495–500
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F. Götze, A. Yu. Zaitsev, “Estimates for the rate of strong approximation in Hilbert space”, Siberian Math. J., 52:4 (2011), 628–638
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A. Yu. Zaitsev, “Optimal estimates for the rate of strong Gaussian approximation in the infinite dimensional invariance principle”, J. Math. Sci. (N. Y.), 188:6 (2013), 689–693
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A. Yu. Zaitsev, “The accuracy of strong Gaussian approximation for sums of independent random vectors”, Russian Math. Surveys, 68:4 (2013), 721–761
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Trapani L., “Comments on: Extensions of Some Classical Methods in Change Point Analysis Discussion”, Test, 23:2 (2014), 283–286
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Berkes I., Liu W., Wu W.B., “Komlos-Major-Tusnady Approximation Under Dependence”, Ann. Probab., 42:2 (2014), 794–817
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Merlevede F., Rio E., “Strong Approximation For Additive Functionals of Geometrically Ergodic Markov Chains”, Electron. J. Probab., 20 (2015), 1–27
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M. A. Lifshits, Ya. Yu. Nikitin, V. V. Petrov, A. Yu. Zaitsev, A. A. Zinger, “Toward the history of the Saint Petersburg school of probability and statistics. I. Limit theorems for sums of independent random variables”, Vestn. St Petersb. Univ. Math., 51:2 (2018), 144–163
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