Abstract
We prove the functional limit theorem, i.e., the invariance principle, for sequences of normalized U- and V-statistics of arbitrary orders with canonical kernels, defined on samples of growing size from a stationary sequence of random variables under the α- or φ-mixing conditions. The corresponding limit stochastic processes are described as polynomial forms of a sequence of dependent Wiener processes with a known covariance.
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References
P. Billingsley, Convergence of ProbabilityMeasures, Willey, N.Y.-Boston, 1968.
I. S. Borisov and A. A. Bystrov, Constructing a stochastic integral of a nonrandom function without orthogonality of the integrating measure, Theory Probab. Appl. (2006), 50(1), 53.
I. S. Borisov and A. A. Bystrov, Limit theorems for canonical von Mises statistics based on dependent observations, Siberian Mathematical Journal, 2006, 47(6), 980.
I. S. Borisov and N. V. Volodko, Orthogonal series and limiti theorems for caninical U- and V-statistics of stationarily connected observations, Siberian Advances in Mathematics, 2008, 18(4), 242.
I. S. Borisov and V. A. Zhechev, A functional limit theorem for canonical U-processes of dependent observation, Siberian Mathematical Journal, 2011, 52(4), 593.
A. A. Borovkov, Probability Theory, Gordon and Breach Science Publishers, Amsterdam, 1998.
H. Dehling, M. Denker, and W. Philipp, The almost sure invariance principle for the empirical process of U-statistic structure, Ann. Inst. H. Poincare Probab. Statist. 1987, 23(2) 121.
M. Denker, C. Grillenberger, and G. Keller, A note on invariance principles for von Mises’ statistics, Metrika, 1985, 32,(3, 4), 197.
P. Doukhan, Mixing. Properties and Examples Lecture Notes in Statistics, 85. New York: Springer-Verlag, 1994.
I. I. Gihman and A. V. Skorohod, Introduction to Theory of Random Processes, “Nauka”, Moscow, 1965 [in Russian]
A. N. Kolmogorov and S. V. Fomin, Elements of Theory of Functions and of Functional Analysis, Second edition, “Nauka”, Moscow, 1968 [Russian].
R. von Mises, On the asymptotic distribution of differentiable statistical functionals, Ann. Math. Statist. 1947, 18, 309.
A. F. Ronzhin, Functional limit theorems for U-statistics//Mathematical Notes, 1986, 40(5), 886.
H. Rubin and R. Vitale, Asymptotic distribution of symmetric statistics, Ann. Statist. 1980, 8(1), 165.
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Original Russian Text © I. S. Borisov and V. A. Zhechev, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 2, pp. 28–44.
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Borisov, I.S., Zhechev, V.A. Invariance principle for canonical U- and V-statistics based on dependent observations. Sib. Adv. Math. 25, 21–32 (2015). https://doi.org/10.3103/S1055134415010034
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DOI: https://doi.org/10.3103/S1055134415010034