Marcinkiewicz-Zygmund Type Law of Large Numbers for Double Arrays of Random Elements in Banach Spaces
ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2009, Vol. 30, No. 4, pp. 337–346. c Pleiades Publishing, Ltd., 2009.
Marcinkiewicz-Zygmund Type Law of Large Numbers for Double
Arrays of Random Elements in Banach Spaces
Le Van Dung1* , Thuntida Ngamkham2 , Nguyen Duy Tien1** , and A. I. Volodin3***
1
Faculty of Mathematics, National University of Hanoi, 3 34 Nguyen Trai, Hanoi, Vietnam
2
3
Department of Mathematics and Statistics, Thammasat University, Rangsit Center,
Pathumthani 12121, Thailand
School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley,
WA 6009, Australia
Received July 30, 2009
Abstract—In this paper we establish Marcinkiewicz-Zygmund type laws of large numbers for double arrays of random elements in Banach spaces. Our results extend those of Hong and
Volodin [6].
2000 Mathematics Subject Classification: 60B11, 60B12, 60F15, 60G42
DOI: 10.1134/S1995080209040118
Key words and phrases: Marcinkiewicz-Zygmund inequality, Rademacher type p Banach spaces, Martingale type p Banach spaces, Double arrays of random elements, Strong and
Lp laws of large numbers.
1. INTRODUCTION
Marcinkiewicz-Zygmund type strong laws of large numbers were studied by many authors. In 1981,
Etemadi [3] proved that if {Xn ; n ≥ 1} is a sequence of pairwise i.i.d. random variables with EX1 < ∞,
1 n
(Xi − EX1 ) = 0 a.s. then lim n i=1
Later, in 1985, Choi and Sung [2] have shown that if {Xn ; n ≥ 1} are pairwise independent and are dominated in distribution by a random variable X with E|X|p (log+ |X|)2 < ∞, 1 < p < 2, then lim 1
n
(Xi − EXi ) = 0 a.s.
1
np
i=1
Recently, Hong and Hwang [5], Hong and Volodin [6] studied Marcinkiewicz-Zygmund strong law of large numbers for double sequence of random variables, Quang and Thanh [12] established the
Marcinkiewicz-Zygmund strong law of large numbers for blockwise adapted sequence. In this paper, we extend the results of Hong
Marcinkiewicz-Zygmund Type Law of Large Numbers for Double
Arrays of Random Elements in Banach Spaces
Le Van Dung1* , Thuntida Ngamkham2 , Nguyen Duy Tien1** , and A. I. Volodin3***
1
Faculty of Mathematics, National University of Hanoi, 3 34 Nguyen Trai, Hanoi, Vietnam
2
3
Department of Mathematics and Statistics, Thammasat University, Rangsit Center,
Pathumthani 12121, Thailand
School of Mathematics and Statistics, University of Western Australia, 35 Stirling Highway, Crawley,
WA 6009, Australia
Received July 30, 2009
Abstract—In this paper we establish Marcinkiewicz-Zygmund type laws of large numbers for double arrays of random elements in Banach spaces. Our results extend those of Hong and
Volodin [6].
2000 Mathematics Subject Classification: 60B11, 60B12, 60F15, 60G42
DOI: 10.1134/S1995080209040118
Key words and phrases: Marcinkiewicz-Zygmund inequality, Rademacher type p Banach spaces, Martingale type p Banach spaces, Double arrays of random elements, Strong and
Lp laws of large numbers.
1. INTRODUCTION
Marcinkiewicz-Zygmund type strong laws of large numbers were studied by many authors. In 1981,
Etemadi [3] proved that if {Xn ; n ≥ 1} is a sequence of pairwise i.i.d. random variables with EX1 < ∞,
1 n
(Xi − EX1 ) = 0 a.s. then lim n i=1
Later, in 1985, Choi and Sung [2] have shown that if {Xn ; n ≥ 1} are pairwise independent and are dominated in distribution by a random variable X with E|X|p (log+ |X|)2 < ∞, 1 < p < 2, then lim 1
n
(Xi − EXi ) = 0 a.s.
1
np
i=1
Recently, Hong and Hwang [5], Hong and Volodin [6] studied Marcinkiewicz-Zygmund strong law of large numbers for double sequence of random variables, Quang and Thanh [12] established the
Marcinkiewicz-Zygmund strong law of large numbers for blockwise adapted sequence. In this paper, we extend the results of Hong
References: 1. Y. S. Chao and H. Teicher, Probability Theory. Independence, Interchangeability, Martingale (Springer, New York, 1997). 3. N. Etemadi, An elementary proof of the strong law of large numbers, Z. Wahrsch. Verw. Gebiete 55 (1), 119 (1981). indexed random variables, J. Multivariate Anal. 86 (2), 398 (2003). (Birkhauser, Boston, 1992). 10. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (3–4), 326 (1975). 11. G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Probability and Analysis (Varenna, 1985); Lecture Notes in Math 14. F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11, 347 (1961).