Summary
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berk, R.H.: Limiting behaviour of posterior distributions when the model is incorrect. Ann. Math. Stat.37, 1966, 51–58.
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.
Bickel, P.J., andM.J. Wichura: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Stat.42, 1971, 1656–1670.
Billingsley, P.: Convergence of probability measures. J. Wiley, New York 1968.
Denker, M., andG. Keller: OnU-statistics and v. Mises' statistics for weakly dependent processes. Z. Wahrscheinlichkeitstheorie verw. Geb.64, 1983, 505–522.
Doob, J.L.: Stochastic processes. J. Wiley, New York 1953.
Dunford, N., andJ.T. Schwarz: Linear Operators. Interscience 1957.
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.
Gregory, C.C.: Large sample theory forU-statistics and tests of fit. Ann. Statistics5, 1977, 110–123.
Hall, P.: The invariance principle forU-statistics. Stoch. Processes and their Appl.9, 1979, 163–174.
Hoeffding, W.: A class of statistics with asymptotically normal distribution. Ann. Math. Stat.19, 1948, 325–346.
Kiefer, J.: Skorohod embedding of multivariate R.V.'s and the sample D.F.. Z. Wahrscheinlichkeitstheorie verw. Geb.24, 1972, 1–35.
Lehmann, E.L.: Consistency and unbiasedness of certain non-parametric tests. Ann. Math. Stat.22, 1951, 165–179.
Loynes, R.M.: An invariance principle for reverse martingales. Proc. Amer. Math. Soc.25, 1970, 56–64.
—: On the weak convergence ofU-statistic processes and of the empirical processes. Math. Proc. Cambridge. Philos. Soc.83, 1978, 269–272.
Miller, R.G., andP.K. Sen: Weak convergence ofU-statistics and v. Mises' differentiable statistical functions. Ann. Math. Stat.43, 1972, 31–41.
v. Mises, R.: On the asymptotic distribution of differentiable statistical functions. Ann. Math. Stat.18, 1947, 309–348.
Müller, D.W.: On Glivenko-Cantelli convergence. Z. Wahrscheinlichkeitstheorie verw. Geb.16, 1970, 195–210.
Neuhaus, G.: On weak convergence of stochastic processes with multi-dimensional time parameter. Ann. Math. Stat.42, 1971, 1285–1295.
—: Functional limit theorems forU-statistics in the degenerate case. J. Multivariate Analysis7, 1977, 424–439.
Sen, P.K.: Weak convergence of generalizedU-statistics. Ann. Prob.2, 1974, 90–102.
—: Almost sure convergence of generalizedU-statistics. Ann. of Prob.5, 1977, 287–290.
Shorack, G.R., andR.T. Smythe: Inequalities for max |S k |/b k wherekεNr. Math. Soc.54, 1976, 331–336.
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.
Author information
Authors and Affiliations
Additional information
The work of the third author has been supported by the Deutsche Forschungsgemeinschaft.
Rights and permissions
About this article
Cite this article
Denker, M., Grillenberger, C. & Keller, G. A note on invariance principles for v. Mises' statistics. Metrika 32, 197–214 (1985). https://doi.org/10.1007/BF01897813
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01897813