ON THE WEAK LAWS WITH RANDOM INDICES FOR PARTIAL SUMS FOR ARRAYS OF RANDOM ELEMENTS IN MARTINGALE TYPE p BANACH SPACES

Title & Authors
ON THE WEAK LAWS WITH RANDOM INDICES FOR PARTIAL SUMS FOR ARRAYS OF RANDOM ELEMENTS IN MARTINGALE TYPE p BANACH SPACES
Sung, Soo-Hak; Hu, Tien-Chung; Volodin, Andrei I.;

Abstract
Sung et al. [13] obtained a WLLN (weak law of large numbers) for the array $\small{\{X_{{ni},\;u_n{\leq}i{\leq}v_n,\;n{\leq}1\}}$ of random variables under a Cesaro type condition, where $\small{\{u_n{\geq}-{\infty},\;n{\geq}1\}}$ and $\small{\{v_n{\leq}+{\infty},\;n{\geq}1\}}$ large two sequences of integers. In this paper, we extend the result of Sung et al. [13] to a martingale type p Banach space
Keywords
arrays of random elements;convergence in probability;martingale 쇼;e p Banach space;weak law of large numbers;randomly indexed sums;martingale difference sequence;Cesaro type condition;
Language
English
Cited by
1.
WEAK LAW OF LARGE NUMBERS FOR ADAPTED DOUBLE ARRAYS OF RANDOM VARIABLES,;;

대한수학회지, 2008. vol.45. 3, pp.795-805
References
1.
A. Adler, A. Rosalsky, and A. Volodin, A mean convergence theorem and weak law for arrays of random elements in martingale type p Banach spaces, Statist. Probab. Lett. 32 (1997), no. 2, 167-174

2.
S. E. Ahmed, S. H. Sung, and A. Volodin, Mean convergence theorem for arrays of random elements in martingale type p Banach spaces, Bull. Inst. Math. Acad. Sinica 30 (2002), no. 2, 89-95

3.
A. Gut, The weak law of large numbers for arrays, Statist. Probab. Lett. 14 (1992), no. 1, 49-52

4.
J. Hoffmann-Jorgensen and G. Pisier, The law of large numbers and the central limit theorem in Banach spaces, Ann. Probability 4 (1976), no. 4, 587-599

5.
D. H. Hong and S. Lee, A general weak law of large numbers for arrays, Bull. Inst. Math. Acad. Sin. 24 (1996), no. 3, 205-209

6.
D. H. Hong and K. S. Oh, On the weak law of large numbers for arrays, Statist. Probab. Lett. 22 (1995), no. 1, 55-57

7.
D. H. Hong, M. Ordonez Cabrera, S. H. Sung, and A. Volodin, On the weak law for randomly indexed partial sums for arrays of random elements in martingale type p Banach spaces, Statist. Probab. Lett. 46 (2000), no. 2, 177-185

8.
P. Kowalski and Z. Rychlik, On the weak law of large numbers for randomly indexed partial sums for arrays, Ann. Univ. Mariae Curie-Sklodowska Sect. A 51 (1997), no. 1, 109-119

9.
G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326-350

10.
G. Pisier, Probabilistic methods in the geometry of Banach spaces, in: G. Lette and M. Pratelli, Eds., Probability and Analysis, Lectures given at the 1st 1985 Session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held at Varenna(Como), Italy, May 31-June 8, 1985, Lecture Notes in Mathematics (Springer-Verlag, Berlin), Vol. 1206 (1986), 167-241

11.
F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374

12.
S. H. Sung, Weak law of large numbers for arrays, Statist. Probab. Lett. 38 (1998), no. 2, 101-105

13.
S. H. Sung, T.-C. Hu, and A. Volodin, On the weak laws for arrays of random variables, Statist. Probab. Lett. 72 (2005), no. 4, 291-298