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The almost sure behaviour of the oscillation modulus of the multivariate empirical process

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  • Einmahl, J.H.J.

    (Tilburg University, School of Economics and Management)

  • Ruymgaart, F.H.

Abstract

Let [omega]n denote the oscillation modulus of the uniform multivariate empirical process, defined as the variation of the process over multi-dimensional intervals with Lebesgue measure not exceeding an [epsilon] (0,1). The a.s. limiting behavior of [omega]n is established for sequences {an} of five different orders of magnitude that constitute an essentially complete spectrum of possibilities. Extensions to processes with underlying d.f. more general than the uniform are indicated. The results may have applications in density estimation and in the theory of multivariate spacings. For related results in the univariate case we refer in particular to Mason, Shorack and Wellner (1983) and Mason (1984), and for a general setting to Alexander (1984).
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Suggested Citation

  • Einmahl, J.H.J. & Ruymgaart, F.H., 1987. "The almost sure behaviour of the oscillation modulus of the multivariate empirical process," Other publications TiSEM 94429b0f-0175-452f-b41b-f, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:94429b0f-0175-452f-b41b-f7312803f9e9
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    Cited by:

    1. Deheuvels, Paul & Peccati, Giovanni & Yor, Marc, 2006. "On quadratic functionals of the Brownian sheet and related processes," Stochastic Processes and their Applications, Elsevier, vol. 116(3), pages 493-538, March.
    2. Peng, Liang & Qi, Yongcheng, 2008. "Bootstrap approximation of tail dependence function," Journal of Multivariate Analysis, Elsevier, vol. 99(8), pages 1807-1824, September.
    3. Gery Geenens & Arthur Charpentier & Davy Paindaveine, 2014. "Probit Transformation for Nonparametric Kernel Estimation of the Copula Density," Working Papers ECARES ECARES 2014-23, ULB -- Universite Libre de Bruxelles.
    4. Alexandre Leblanc, 2009. "Chung–Smirnov property for Bernstein estimators of distribution functions," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 21(2), pages 133-142.
    5. John H. J. Einmahl & Andrew Rosalsky, 2001. "The Functional Law of the Iterated Logarithm for the Empirical Process Based on Sample Means," Journal of Theoretical Probability, Springer, vol. 14(2), pages 577-597, April.

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