Weak convergence of the weighted multiparameter empirical process

  • Ludger Rüschendorf
Conference paper
Part of the Lecture Notes in Mathematics book series (LNM, volume 821)


By means of the Poisson type representation of multivariate empirical processes and using a generalization of the Birnbaum Marshall inequality it is shown that the empirical process converges in distribution even when it is weighted by some unbounded weighting functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Bickel, P.J., Wichura, M.J. (1971) Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42, 1656–1670.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Chibisov, D.M. (1965) An investigation of the asymptotic power of tests of fit. Theor. Prob. and Appl. 10, 421–437.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Govindarajulu, Z., Le Cam, L., Raghavachari, M. (1967) Generalizations of theorems of Chernoff-Savage on asymptotic normality of nonparametric test statistics. Proc. of Fifth Berkeley Symp. on Math. Statist. and Prob. 609–638, Univ. of Calif. Press.Google Scholar
  4. [4]
    Kac, M. (1949) On deviations between theoretical and empirical distributions. Proc. Nat. Acad. Sci. 35, 252–257.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Neuhaus, G. (1971) On weak convergence of stochastic processes with multidimensional time parameter. Ann. Math. Statist. 42, 1285–1295.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Pyke, R., Shorack, G.R. (1968) Weak convergence of a two-sample empirical process and a new approach to Chernoff-Savage Theorems. Ann. Math. Statist. 39, 755–771.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Rosenblatt, M. (1975) A quadratic measure of deviation of two-dimensional density estimates and a test of independence. Ann. Statist. 3, 1–14.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Rüschendorf, L. (1976) Asymptotic distributions of multivariate rank order statistics. Ann. Statist. 3, 912–923.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Wichura, M. (1973) Some Strassen-type laws of the iterated logarithm for multi-parameter stochastic processes with independent increments. Ann. Probab. 1, 272–296.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Ludger Rüschendorf
    • 1
  1. 1.Institut für Statistik und Wirtschaftsmathematik, TH-AachenAachenGermany

Personalised recommendations