ON THE RATE OF COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF ARRAYS OF RANDOM ELEMENTS

Title & Authors
ON THE RATE OF COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF ARRAYS OF RANDOM ELEMENTS
Sung, Soo-Hak; Volodin Andrei I.;

Abstract
Let {$\small{V_{nk},\;k\;{\geq}\;1,\;{\geq}\;1}$} be an array of rowwise independent random elements which are stochastically dominated by a random variable X with $E\|X\|^{\frac{\alpha}{\gamma}+{\theta}}log^{\rho}(\|X\|)\;<\;{\infty}$ for some ${\rho}\;>\;0,\;{\alpha}\;>\;0,\;{\gamma}\;>\;0,\;{\theta}\;>\;0$ such that ${\theta}+{\alpha}/{\gamma}<2$. Let {$\small{a_{nk},k{\geq}1,n{\geq}1}$) be an array of suitable constants. A complete convergence result is obtained for the weighted sums of the form $\small{\sum{^\infty_k_=_1}\;a_{nk}V_{nk}}$.
Keywords
arrays of random elements;rowwise independence;weighted sums;complete convergence;rate of convergence;convergence in probability;
Language
English
Cited by
1.
COMPLETE CONVERGENCE FOR WEIGHTED SUMS OF RANDOM ELEMENTS,;

대한수학회보, 2010. vol.47. 2, pp.369-383
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References
1.
S. E. Ahmed, R. G. Antonini, and A. I. Volodin, On the rate of complete conver- gence for weighted sums of arrays of Banach space valued random elements with application to moving average processes, Statist. Probab. Lett. 58 (2002), no. 2, 185-194

2.
L. E. Baum and M. Katz, Convergence rates in the law of large numbers, Trans. Amer. Math. Soc. 120 (1965), 108-123

3.
P. Erdos, On a theorem of Hsu and Robbins, Ann. Math. Statistics 20 (1949), 286-291

4.
P. Erdos, Remark on my paper 'On a theorem of Hsu and Robbins', Ann. Math. Statistics 21 (1950), 138

5.
P. L. Hsu and H. Robbins, Complete convergence and the law of large numbers, Proc. Nat. Acad. Sci. U.S.A. 33 (1947), 25-31

6.
T.-C. Hu, D. Li, A. Rosalsky, and A. I. Volodin, On the rate of complete con- vergence for weighted sums of arrays of Banach space valued random elements, Theory Probab. Appl. 47 (2003), 455-468

7.
T.-C. Hu, A. Rosalsky, D. Szynal and A. I. Volodin, On complete convergence for arrays of rowwise independent random elements in Banach spaces, Stochastic Anal. Appl. 17 (1999), no. 6, 963-992

8.
A. Kuczmaszewska and D. Szynal, On complete convergence in a Banach space, Internat. J. Math. Math. Sci. 17 (1994), no. 1, 1-14

9.
S. H. Sung, Complete convergence for weighted sums of arrays of rowwise in- dependent B-valued random variables, Stochastic Anal. Appl. 15 (1997), no. 2, 255-267

10.
A. Volodin, R. G. Antonini and T.-C. Hu, A note on the rate of complete con- vergence for weighted sums of arrays of Banach space valued random elements, Lobachevskii J. Math. (electronic) 15 (2004), 21-33

11.
X. Wang, M. B. Rao and X. Yang, Convergence rates on strong laws of large numbers for arrays of rowwise independent elements, Stochastic Anal. Appl. 11 (1993), no. 1, 115-132