The Complete Convergence for Sequence of Arbitrary Random Variables in Hilbert Space and its Application
Main Article Content
Abstract
In this work, we establish the complete convergence for sequences of arbitrary random variables taking values in Hilbert space with general normalizing sequences. As corollaries, we present some convergence results for -valued martingale difference sequences. Finally, the complete convergence of degenerate von Mises statistics is investigated.
Keywords:
Complete convergence, Martingale difference, Hilbert space.
References
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W. Yang, X. Yang, A Note on Strong Limit Theorems for Arbitrary Stochastic Sequences, Statistics & Probability Letters, Vol. 78, No.14, 2008, pp. 2018-2023, https://doi.org/10.1016/j.spl.2008.01.084.
H. Zhang, R. Hao, W. Yang, Z. Bao, Some Limit Theorems for the Log-optimal Portfolio, Journal of Inequalities and Applications, Vol. 310, 2014, pp. 1-10, https://doi.org/10.1186/1029-242X-2014-310.
Z. Wang, Y. Dong, F. Ding, On Almost Sure Convergence for Sums of Stochastic Sequence, Communications in Statistics-Theory anD Methods, Vol. 48, No. 14, 2019, pp. 3609-3621, https://doi.org/10.1080/03610926.2018.1478102.
D. Qiu, P. Chen, V. Andrei, Complete Moment Convergence for L^P-mixingales, Acta Mathematica Scientia,
Vol. 37, No. 5, 2017, pp. 1319-1330, https://doi.org/10.1016/S0252-9602(17)30075-9.
S. Gan, Rosenthal’s Inequality and Its Applications for B-valued Random Elements, Acta Math Sci, Vol. 30, No. 2, 2010, pp. 327-334.
F. Cirstea, V. D. Radulescu, Nonlinear Problems with Boundary Blow-up: a Karamata Regular Variation Theory Approach, Asymptotic Analysis, Vol. 46, No. 3, 2006, pp. 275-298, https://doi.org/10.3233/ASY-2006-732.
N. H. Bingham, C. M. Goldie, J. L. Teugels, Regular Variation, Encyclopedia of Mathematics and Its Applications, Series 27, Cambridge University Press, Cambridge, 1987.
R. Bojanic, E. Seneta, Slowly Varying Functions and Asymptotic Relations, J. Math. Anal. Appl, Vol. 34, 1971, pp. 302-315, https://doi.org/10.1016/0022-247X(71)90114-4.
H. Sun, Mercer Theorem for RKHS on Noncompact Sets, Journal of Complexity, Vol. 21, No. 3, 2005,
pp. 337-349, https://doi.org/10.1016/j.jco.2004.09.002.
H. Dehling, O. S. Sharipov, M. Wendler, Bootstrap for Dependent Hilbert Space-valued Random Variables with Application to von Mises Statistics, J. Multivariate Anal, Vol. 133C, 2015, pp. 200-215, https://doi.org/10.1016/j.jmva.2014.09.011.
T. C. Son, T. M. Cuong, L. V. Dung, On the Almost Sure Convergence for Sums of Negatively Superadditive Dependent Random Vectors in Hilbert Spaces and Its Application, Commun. Statist. Theor. Methods, Vol. 49, No. 11, 2020, pp. 1-17, https://doi.org/10.1080/03610926.2019.1584304.
T. C. Son, T. M. Cuong, L. V. Dung, T. V. Chien, On the Convergence for Weighted Sums of Hilbert-valued Coordinatewise Pairwise NQD Random Variables and Its Application, Commun. Statist. Theor. Methods,
Vol. 52, No. 23, 2022, pp. 8371-8387, https://doi.org/10.1080/03610926.2022.2062604.