, Volume 32, Issue 1, pp 197–214 | Cite as

A note on invariance principles for v. Mises' statistics

  • M. Denker
  • C. Grillenberger
  • G. Keller


We extend Filippova's result on weak convergence of v. Mises' functionals and prove a weak invariance principle. Applications toU-statistics are given and extensions to contiguity and weakly dependent processes are briefly discussed.


Stochastic Process Probability Theory Economic Theory Weak Convergence Invariance Principle 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Berk, R.H.: Limiting behaviour of posterior distributions when the model is incorrect. Ann. Math. Stat.37, 1966, 51–58.Google Scholar
  2. Berkes, I., andW. Philipp: Almost sure invariance principle for the empirical distribution function of mixing random variables. Z. Wahrscheinlichkeitstheorie verw. Geb.41, 1977, 115–137.Google Scholar
  3. Bickel, P.J., andM.J. Wichura: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat.42, 1971, 1656–1670.Google Scholar
  4. Billingsley, P.: Convergence of probability measures. J. Wiley, New York 1968.Google Scholar
  5. Denker, M., andG. Keller: OnU-statistics and v. Mises' statistics for weakly dependent processes. Z. Wahrscheinlichkeitstheorie verw. Geb.64, 1983, 505–522.Google Scholar
  6. Doob, J.L.: Stochastic processes. J. Wiley, New York 1953.Google Scholar
  7. Dunford, N., andJ.T. Schwarz: Linear Operators. Interscience 1957.Google Scholar
  8. Filippova, A.A.: Mises' theorem on the asymptotic behaviour of functionals of empirical distribution functions and its statistical applications. Theory of Prob. and its Appl.VII, 1, 1962, 24–57.Google Scholar
  9. Gregory, C.C.: Large sample theory forU-statistics and tests of fit. Ann. Statistics5, 1977, 110–123.Google Scholar
  10. Hall, P.: The invariance principle forU-statistics. Stoch. Processes and their Appl.9, 1979, 163–174.Google Scholar
  11. Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat.19, 1948, 325–346.Google Scholar
  12. Kiefer, J.: Skorohod embedding of multivariate R.V.'s and the sample D.F.. Z. Wahrscheinlichkeitstheorie verw. Geb.24, 1972, 1–35.Google Scholar
  13. Lehmann, E.L.: Consistency and unbiasedness of certain non-parametric tests. Ann. Math. Stat.22, 1951, 165–179.Google Scholar
  14. Loynes, R.M.: An invariance principle for reverse martingales. Proc. Amer. Math. Soc.25, 1970, 56–64.Google Scholar
  15. —: On the weak convergence ofU-statistic processes and of the empirical processes. Math. Proc. Cambridge. Philos. Soc.83, 1978, 269–272.Google Scholar
  16. Miller, R.G., andP.K. Sen: Weak convergence ofU-statistics and v. Mises' differentiable statistical functions. Ann. Math. Stat.43, 1972, 31–41.Google Scholar
  17. v. Mises, R.: On the asymptotic distribution of differentiable statistical functions. Ann. Math. Stat.18, 1947, 309–348.Google Scholar
  18. Müller, D.W.: On Glivenko-Cantelli convergence. Z. Wahrscheinlichkeitstheorie verw. Geb.16, 1970, 195–210.Google Scholar
  19. Neuhaus, G.: On weak convergence of stochastic processes with multi-dimensional time parameter. Ann. Math. Stat.42, 1971, 1285–1295.Google Scholar
  20. —: Functional limit theorems forU-statistics in the degenerate case. J. Multivariate Analysis7, 1977, 424–439.Google Scholar
  21. Sen, P.K.: Weak convergence of generalizedU-statistics. Ann. Prob.2, 1974, 90–102.Google Scholar
  22. —: Almost sure convergence of generalizedU-statistics. Ann. of Prob.5, 1977, 287–290.Google Scholar
  23. Shorack, G.R., andR.T. Smythe: Inequalities for max |S k |/b k wherekεNr. Math. Soc.54, 1976, 331–336.Google Scholar
  24. Wichura, M.J.: Inequalities with applications to the weak convergence of random processes with multi-dimensional time parameters. Ann. Math. Stat.40, 1969, 681–687.Google Scholar

Copyright information

© Physica-Verlag Ges.m.b.H. 1985

Authors and Affiliations

  • M. Denker
    • 1
  • C. Grillenberger
    • 2
  • G. Keller
    • 3
  1. 1.Universität GöttingenGöttingen
  2. 2.Gesamthochschule KasselKassel
  3. 3.Universität HeidelbergHeidelberg

Personalised recommendations