Browse > Article
http://dx.doi.org/10.4134/JKMS.2012.49.5.1053

CONVERGENCE OF DOUBLE SERIES OF RANDOM ELEMENTS IN BANACH SPACES  

Tien, Nguyen Duy (Faculty of Mathematics National University of Hanoi)
Dung, Le Van (Faculty of Mathematics Danang University of Education)
Publication Information
Journal of the Korean Mathematical Society / v.49, no.5, 2012 , pp. 1053-1064 More about this Journal
Abstract
For a double array of random elements $\{X_{mn};m{\geq}1,n{\geq}1\}$ in a $p$-uniformly smooth Banach space, $\{b_{mn};m{\geq}1,n{\geq}1\}$ is an array of positive numbers, convergence of double random series ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}X_{mn}$, ${\sum}^{\infty}_{m=1}{\sum}^{\infty}_{n=1}b^{-1}_{mn}X_{mn}$ and strong law of large numbers $$b^{-1}_{mn}\sum^m_{i=1}\sum^n_{j=1}X_{ij}{\rightarrow}0$$ as $$m{\wedge}n{\rightarrow}{\infty}$$ are established.
Keywords
convergence of double random series; strong laws of large numbers; $p$-uniformly smooth Banach spaces; double array of random elements;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 N. D. Tien, On Kolmogorov's three series theorem and mean square convergence of martingales in a Banach space, Theory Probab. Appl. 24 (1980), no. 4, 797-808.   DOI
2 W. A. Woyczynski, Geometry and martingales in Banach spaces, ProbabilityWinter School (Proc. Fourth Winter School, Karpacz, 1975), pp. 229-275. Lecture Notes in Math., Vol. 472, Springer, Berlin, 1975.
3 S. Gan, On almost sure convergence of weighted sums of random element sequences, Acta Math. Sci. Ser. B Engl. Ed. 30 (2010), no. 4, 1021-1028.
4 J. I. Hong and J. Tsay, A strong law of large numbers for random elements in Banach spaces, Southeast Asian Bull. Math. 34 (2010), no. 2, 257-264.
5 D. Landers and L. Rogge, Laws of large numbers for pairwise independent uniformly integrable random variables, Math. Nachr. 130 (1987), 189-192.   DOI
6 M. Ordonez Cabrera, Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (1994), no. 2, 121-132.
7 G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and analysis (Varenna, 1985), 167241, Lecture Notes in Math., 1206, Springer, Berlin, 1986.
8 F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374.   DOI
9 N. V. Quang, L. V. Thanh, and N. D. Tien, Almost sure convergence for double arrays of block-wise M-dependent random elements in Banach spaces, Georgian Mathematical Journal 18 (2011), 777-800.   DOI
10 A. Rosalsky and L. V. Thanh, Strong and weak laws of large numbers for double sums of independent random elements in Rademacher type p Banach spaces, Stoch. Anal. Appl. 24 (2006), no. 6, 1097-1117.   DOI   ScienceOn
11 U. Stadtmulle and L. V. Thanh, On the strong limit theorems for double arrays of blockwise M-dependent random variables, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 10, 19231934.
12 C. Su and T. J. Tong, Almost sure convergence of the general Jamison weighted sum of B-valued random variables, Acta Math. Sin. (Engl. Ser.) 20 (2004), no. 1, 181-192.   DOI
13 L. V. Dung and N. D. Tien, Mean convergence theorems and weak laws of large numbers for double arrays of random elements in Banach spaces, Bull. Korean Math. Soc. 47 (2010), no. 3, 467{482.   과학기술학회마을
14 L. V. Dung, Th. Ngamkham, N. D. Tien and A. I. Volodin, Marcinkiewicz-Zygmund type law of large numbers for double arrays of random elements in Banach spaces, Lobachevskii J. Math. 30 (2009), no. 4, 337-346.   DOI