Abstract
We establish exact rates of convergence in functional limit theorems forU-statistic type processes in the sense of Prokhorov's metric. It turns out that the rates are the same as for the classical partial sum process. The results can be used to derive statements of Berry-Esséen type and of Strassen type, respectively, for changepoint estimators.
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References
Arak, T. V. (1975). On an estimate of A. A. Borovkov.Theor. Probability Appl. 20, 372–373.
Borovkov, A. A. (1973). On the rate of convergence for the invariance principle.Theor. Probability Appl. 18, 207–225.
Chung, K. L. (1974).A Course in Probability Theory, Academic Press, Orlando, Florida.
Csörgő, M., and Horváth, L. (1988). Invariance principles for changepoint problems.J. Multivariate Anal. 27, 151–168.
Csörgő, M., and Révész P. (1981).Strong Approximations in Probability and Statistics, Academic Press, New York, San Francisco, London.
Ferger, D., and Stute, W. (1992). Convergence of changepoint estimators.Stochastic Process. Appl. 42, 345–351.
Häusler, E. (1984). An Exact Rate of Convergence in the Functional Central Limit Theorem for Special Martingale Difference Arrays.Z. Wahrsch. verw. Gebiete 65, 523–534.
Hoeffding, W. (1963). Probability inequalities for sums of bounded random variables.J. Amer. Statist. Assoc. 58, 13–30.
Komlós, J., Major, P., and Tusnády, G. (1974). Weak convergence and embedding. In Révész, P. (ed.),Colloquia Mathematica Societatis János Bolyai, 11. Limit Theorems of Probability Theory, North-Holland, Amsterdam-London, pp. 149–165.
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Ferger, D. On exact rates of convergence in functional limit theorems forU-statistic type processes. J Theor Probab 7, 709–723 (1994). https://doi.org/10.1007/BF02214367
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DOI: https://doi.org/10.1007/BF02214367