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Extensions of results of Komlós, Major, and Tusnády to the multivariate case

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  • Einmahl, Uwe

Abstract

A construction method is developed which enables us to establish the well-known approximation results of Komlós, Major, Tusnády (Z. Wahrsch. Verw. Gebiete34 33-58) in the multidimensional setting. The basic tool is a large deviation theorem for conditional distribution functions which may be of independent interest.

Suggested Citation

  • Einmahl, Uwe, 1989. "Extensions of results of Komlós, Major, and Tusnády to the multivariate case," Journal of Multivariate Analysis, Elsevier, vol. 28(1), pages 20-68, January.
    • Handle: RePEc:eee:jmvana:v:28:y:1989:i:1:p:20-68
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    Cited by:

    1. Liu, Weidong & Lin, Zhengyan, 2009. "Strong approximation for a class of stationary processes," Stochastic Processes and their Applications, Elsevier, vol. 119(1), pages 249-280, January.
    2. Joon Y. Park, 2003. "Bootstrap Unit Root Tests," Econometrica, Econometric Society, vol. 71(6), pages 1845-1895, November.
    3. Jingjia Liu & Quirin Vogel, 2021. "Large Deviations of the Range of the Planar Random Walk on the Scale of the Mean," Journal of Theoretical Probability, Springer, vol. 34(4), pages 2315-2345, December.
    4. M. Kayid & M. Shafaei Noughabi & A. M. Abouammoh, 2020. "A Nonparametric Estimator of Bivariate Quantile Residual Life Model with Application to Tumor Recurrence Data Set," Journal of Classification, Springer;The Classification Society, vol. 37(1), pages 237-253, April.
    5. Arup Bose & Rajat Subhra Hazra & Koushik Saha, 2011. "Spectral Norm of Circulant-Type Matrices," Journal of Theoretical Probability, Springer, vol. 24(2), pages 479-516, June.
    6. Vogel, Quirin, 2021. "A note on the intersections of two random walks in two dimensions," Statistics & Probability Letters, Elsevier, vol. 178(C).
    7. Cao, Guanqun & Wang, Li, 2018. "Simultaneous inference for the mean of repeated functional data," Journal of Multivariate Analysis, Elsevier, vol. 165(C), pages 279-295.
    8. S{o}ren Johansen & Morten {O}rregaard Nielsen, 2022. "Weak convergence to derivatives of fractional Brownian motion," Papers 2208.02516, arXiv.org, revised Oct 2022.
    9. Marinucci, D. & Robinson, P. M., 2000. "Weak convergence of multivariate fractional processes," Stochastic Processes and their Applications, Elsevier, vol. 86(1), pages 103-120, March.
    10. Yuzo Hosoya, 2005. "Fractional Invariance Principle," Journal of Time Series Analysis, Wiley Blackwell, vol. 26(3), pages 463-486, May.
    11. Csörgo, Miklós & Horváth, Lajos, 1996. "A note on the change-point problem for angular data," Statistics & Probability Letters, Elsevier, vol. 27(1), pages 61-65, March.
    12. Amir, Gideon & Benjamini, Itai & Gurel-Gurevich, Ori & Kozma, Gady, 2020. "Random walk in changing environment," Stochastic Processes and their Applications, Elsevier, vol. 130(12), pages 7463-7482.
    13. Jirak, Moritz, 2013. "A Darling–Erdös type result for stationary ellipsoids," Stochastic Processes and their Applications, Elsevier, vol. 123(6), pages 1922-1946.
    14. Csörgo, Miklós & Norvaisa, Rimas & Szyszkowicz, Barbara, 1999. "Convergence of weighted partial sums when the limiting distribution is not necessarily Radon," Stochastic Processes and their Applications, Elsevier, vol. 81(1), pages 81-101, May.
    15. Koval, Valery & Schwabe, Rainer, 2003. "A law of the iterated logarithm for stochastic approximation procedures in d-dimensional Euclidean space," Stochastic Processes and their Applications, Elsevier, vol. 105(2), pages 299-313, June.
    16. Park, Joon Y. & Shin, Kwanho & Whang, Yoon-Jae, 2010. "A semiparametric cointegrating regression: Investigating the effects of age distributions on consumption and saving," Journal of Econometrics, Elsevier, vol. 157(1), pages 165-178, July.
    17. Bashtova, Elena & Shashkin, Alexey, 2022. "Strong Gaussian approximation for cumulative processes," Stochastic Processes and their Applications, Elsevier, vol. 150(C), pages 1-18.

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