tailieunhanh - Báo cáo hóa học: " Research Article On the Strong Laws for Weighted Sums "

Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article On the Strong Laws for Weighted Sums | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011 Article ID 157816 8 pages doi 2011 157816 Research Article On the Strong Laws for Weighted Sums of p -Mixing Random Variables Xing-Cai Zhou 1 2 Chang-Chun Tan 3 and Jin-Guan Lin1 1 Department of Mathematics Southeast University Nanjing 210096 China 2 Department of Mathematics and Computer Science Tongling University Tongling Anhui 244000 China 3 School of Mathematics Heifei University of Technology Hefei Anhui 230009 China Correspondence should be addressed to Chang-Chun Tan cctan@ Received 26 October 2010 Revised 5 January 2011 Accepted 27 January 2011 Academic Editor Matti K. Vuorinen Copyright 2011 Xing-Cai Zhou et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Complete convergence is studied for linear statistics that are weighted sums of identically distributed p -mixing random variables under a suitable moment condition. The results obtained generalize and complement some earlier results. A Marcinkiewicz-Zygmund-type strong law is also obtained. 1. Introduction Suppose that Xn n 1 is a sequence of random variables and S is a subset of the natural number set N. Let FS Ơ Xi i e S pn sup corr f g VS X T c N X N dist S T n Vf e L2 Fs g e L2 Ft Ị where n - Cov f Xi i e S g Xj j e T corr f g Ị _ _ . Var f Xi i e S Var g Xj j e T 1 2 Definition . A random variable sequence Xn n 1 is said to be a p -mixing random variable sequence if there exists k e N such that pk 1. 2 Journal of Inequalities and Applications The notion of p -mixing seems to be similar to the notion of p-mixing but they are quite different from each other. Many useful results have been obtained for p -mixing random variables. For example Bradley 1 has established the central limit theorem Byrc and Smolenski 2 and .