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ISSN 1995-0802, Lobachevskii Diary of Mathematics, 2009, Volume. 30, No . 4, pp. 337–346. c Pleiades Submitting, Ltd., 2009.
Marcinkiewicz-Zygmund Type Law of enormous Numbers for Double
Arrays of Randomly Elements in Banach Spaces
Le Vehicle Dung1*, Thuntida Ngamkham2, Nguyen Duy Tien1**, and A. I. Volodin3*** 1
Teachers of Math concepts, National School of Hanoi, 3 34 Nguyen Trai, Hanoi, Vietnam 2
three or more
Department of Mathematics and Statistics, Thammasat University, Rangsit Center, Pathumthani 12121, Asia
School of Mathematics and Statistics, University or college of American Australia, 35 Stirling Road, Crawley, CALIFORNIA 6009, Quotes
Received July 30, 2009
Abstract—In this paper we establish Marcinkiewicz-Zygmund type laws of large quantities for double arrays of random components in Banach spaces. Each of our results prolong those of Hong and Volodin .
2000 Math Subject Classiﬁcation: 60B11, 60B12, 60F15, 60G42 DOI: 12. 1134/S1995080209040118
Key phrases and phrases: Marcinkiewicz-Zygmund inequality, Rademacher type s Banach spaces, Martingale type p Banach spaces, Dual arrays of random elements, Strong and Lp regulations of large amounts.
1 . INTRO
Marcinkiewicz-Zygmund type strong regulations of large amounts were studied by many writers. In 81, Etemadi  proved that if Xn ; n ≥ 1 is a sequence of pairwise i. i. d. unique variables with EX1 < ∞, 1 n
(Xi − EX1 ) sama dengan 0 a. s.
Afterwards, in 85, Choi and Sung  have shown that if Xn ; n ≥ 1 are pairwise independent and they are dominated in distribution by a random changing X with E|X|p (log+ |X|)2 < ∞, you < g < 2, then lim
(Xi − EXi ) = 0 a. t.
Recently, Hong and Hwang , Hong and Volodin  researched Marcinkiewicz-Zygmund solid law of enormous numbers for double series of randomly variables, Quang and Thanh  established the Marcinkiewicz-Zygmund strong regulation of large numbers for blockwise adapted series. In this daily news, we lengthen the results of Hong and Volodin  to some special school of Banach spaces, so-called Banach areas that fulfill the maximal Marcinkiewicz-Zygmund inequality with exponent l (see the deﬁnition below). This category includes Rademacher type g and martingale type s Banach areas, 0 < p ≤ 2 . For the, b ∈ R, greatest extent a, b will probably be denoted by a ∨ w. Throughout this paper, the symbol C will denote a universal constant (0 < C < ∞) which is not automatically the same one out of each overall look. *
E-mail: [email protected] com
E-mail: [email protected] ac. vn
E-mail address: [email protected] uwa. edu. au
DUNG et al.
2 . PRELIMINARIES
Technological deﬁnitions highly relevant to the current job will be mentioned in this section. The Banach space By is said to be of Rademacher type p (1 ≤ g ≤ 2) if there exists a constant C < ∞ such that
for all those independent Times -valued unique elements V1,..., Vn with mean 0. ´
All of us refer you to Pisier  and Woyczynski  for a detailed discussion of this kind of notion. Scalora  released the idea of the conditional expectation of a unique element in a Banach space. For a randomly element Versus and subwoofer σ-algebra G of Farrenheit, the conditional expectation E(V |G) is deﬁned analogously to that inside the random changing case and enjoys similar properties. A genuine separable Banach space Back button is said to be martingale type s (1 ≤ p ≤ 2) if perhaps there exists a ﬁnite positive regular C in a way that for all martingales Sn ; n ≥ 1 with values in X, ∞
sup E||Sn || ≤ C
E||Sn − Sn−1 ||p.
It can be displayed using traditional methods from martingale theory that if X is of martingale type p, after that for all 1 ≤ r < ∞ there exists a ﬁnite constant C such that r
E sup ||Sn ||r ≤ CE
||Sn − Sn−1 ||p
Clearly just about every real separable Banach space is of martingale type 1 and the genuine line (the same as virtually any Hilbert space) is of martingale type 2 . If a real separable Banach space of martingale type p for a few 1 < p...
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New York, 1997).
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indexed random variables, L. Multivariate Anal. 86 (2), 398 (2003).
(Birkhauser, Boston, 1992).
12. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Mathematics. 20 (3–4), 326 (1975).
11. G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: Likelihood and Research (Varenna,
1985); Lecture Notes in Math
18. F. S. Scalora, Abstract martingale affluence theorems, Paciﬁc J. Mathematics. 11, 347 (1961).