Abstract
ForU-statistics taking values in a Hilbert space, we obtain estimates of the rate of convergence in the central limit theorem.
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References
Yu. V. Borovskikh,Theory of U-Statistics a Hilbert Space [in Russian], Preprint 86.78, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1986).
V. S. Korolyuk and Yu. V. Borovskikh,Theory of U-Statistics [in Russian], Naukova Dumka, Kiev (1989).
V. S. Korolyuk and Yu. V. Borovskikh, “Convergence rate in the central limit theorem forUH-statistics,”Ukr. Mat. Zh.,42, No. 8, 1043–1050 (1990).
T. L. Malevich and B. Abdalimov, “Large deviation probabilities forU-statistics,”Teor. Ver. Primen.,24, No. 1, 215–220 (1979).
A. K. Aleshkyavichene, “Large deviation probabilities forU-statistics and Mises functionals,”Teor. Ver. Primen.,35, No. 1, 3–14 (1990).
A. K. Aleshkyavichene, “Large deviations forM-andL-statistics,”Teor. Ver. Primen.,36, No. 4, 774–775 (1991).
L. Saulis and V. Statulyavichus,Limit Theorems for Large Deviations [in Russian], Mokslas, Vilnius (1989).
R. Dasgupta, “On large deviation probability ofU-statistics in non i.i. d. case,”Sankhya. Ser. A,46, No. 1, 110–116 (1984).
P. Y. Bickel, “Edgeworth expansions in nonparametric statistics,”Ann. Statist.,2, No. 1, 1–20 (1974).
M. Vandemaele, “On large deviation probabilities forU-statistics,”Teor. Ver. Primen.,27, No. 3, 573–574 (1982).
M. Vandemaele and N. Veraverbeke, “Cramer type large deviations for linear combinations of order statistics,”Ann. Probab.,10, No. 2, 423–434 (1982).
M. Vandemaele and N. Veraverbeke, “Cramer type large deviations for studentizedU-statistics,”Metrika,32, Nos. 3–4, 165–180 (1985).
W. Hoeffding, “Probabilities inequalities for sums of bounded random variables,”J. Am. Statist. Assoc.,58, No. 301, 13–30 (1963).
H. Rubin and I. Scthuraman, “Probabilities of moderate deviations,”Sankhya. Ser. A,27, No. 2–4, 325–346 (1965).
M. Ghosh, “Probabilities of moderate deviations underm-dependence,”Can. J. Statist. Sect. A, B,2, No. 2, 157–168 (1974).
B. A. Zalesskii, “Large deviation probabilities in a Hilbert space,”Teor. Ver. Primen.,34, No. 4, 650–655 (1989).
B. A. Zalesskii, “Accuracy of the Gaussian approximation in Banach spaces,”Teor. Ver. Primen.,34, No. 4, 815–817 (1989).
V. Bentkus, “On large deviations in Banach spaces,”Teor. Ver. Primen.,31, No. 4, 710–716 (1986).
A. Rachkauskas, “Large deviation probabilities in Linnik zones in a Hilbert space,”Litov. Mat. Sb. (Liet. Mat. Rinkinys),28, No. 3, 520–533 (1988).
V. V. Yurinskii, “On the asymptotics of large deviations in a Hilbert space. I, II,”Teor. Ver. Primen.,36, No. 1, 78–92 (1991).
V. I. Bentkus and A. I. Rachkauskas, “On probabilities of large deviations in Banach spaces,”Probab. Theory Relat. Fields,86, No. 2, 131–154 (1990).
V. S. Korolyuk and Yu. V. Borovskikh,Martingale Approximation [in Russian], Naukova Dumka, Kiev (1988).
D. L. Burkholder, “Distribution function inequalities for martingales,”Ann. Probab.,1, No. 1, 19–42 (1973).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1611–1620, December, 1994.
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Borovskikh, Y.V. Large deviation probabilities forUH-statistics. Ukr Math J 46, 1783–1794 (1994). https://doi.org/10.1007/BF01063167
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DOI: https://doi.org/10.1007/BF01063167