# MEAN CONVERGENCE THEOREMS AND WEAK LAWS OF LARGE NUMBERS FOR DOUBLE ARRAYS OF RANDOM ELEMENTS IN BANACH SPACES

• Dung, Le Van (Faculty of Mathematics Danang University of Education) ;
• Tien, Nguyen Duy (Faculty of Mathematics National University of Hanoi)
• Published : 2010.05.31
• 94 6

#### Abstract

For a double array of random elements {$V_{mn};m{\geq}1,\;n{\geq}1$} in a real separable Banach space, some mean convergence theorems and weak laws of large numbers are established. For the mean convergence results, conditions are provided under which $k_{mn}^{-\frac{1}{r}}\sum{{u_m}\atop{i=1}}\sum{{u_n}\atop{i=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in $L_r$ (0 < r < 2). The weak law results provide conditions for $k_{mn}^{-\frac{1}{r}}\sum{{T_m}\atop{i=1}}\sum{{\tau}_n\atop{j=1}}(V_{ij}-E(V_{ij}|F_{ij})){\rightarrow}0$ in probability where {$T_m;m\;{\geq}1$} and {${\tau}_n;n\;{\geq}1$} are sequences of positive integer-valued random variables, {$k_{mn};m{{\geq}}1,\;n{\geq}1$} is an array of positive integers. The sharpness of the results is illustrated by examples.

#### Keywords

martingale type p Banach spaces;double arrays of random elements;weighted double sums;weak laws of large numbers;mean convergence theorem

#### References

1. A. Adler, A. Rosalsky, and A. I. 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. https://doi.org/10.1016/S0167-7152(97)85593-9
2. T. K. Chandra, Uniform integrability in the Cesaro sense and the weak law of large numbers, SankhyaSer. A 51 (1989), no. 3, 309-317.
3. 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. https://doi.org/10.1214/aop/1176996029
4. D. Landers and L. Rogge, Laws of large numbers for pairwise independent uniformly integrable random variables, Math. Nachr. 130 (1987), 189-192. https://doi.org/10.1002/mana.19871300117
5. M. Ordonez Cabrera, Convergence of weighted sums of random variables and uniform integrability concerning the weights, Collect. Math. 45 (1994), no. 2, 121-132.
6. G. Pisier, Martingales with values in uniformly convex spaces, Israel J. Math. 20 (1975), no. 3-4, 326-350. https://doi.org/10.1007/BF02760337
7. G. Pisier, Probabilistic methods in the geometry of Banach spaces, Probability and analysis (Varenna, 1985), 167-241, Lecture Notes in Math., 1206, Springer, Berlin, 1986. https://doi.org/10.1007/BFb0076302
8. F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374. https://doi.org/10.2140/pjm.1961.11.347
9. S. H. Sung, Weak law of large numbers for arrays of random variables, Statist. Probab. Lett. 42 (1999), no. 3, 293-298. https://doi.org/10.1016/S0167-7152(98)00219-3
10. L. V. Thanh, Mean convergence theorems and weak laws of large numbers for double arrays of random variables, J. Appl. Math. Stoch. Anal. 2006 (2006), Art. ID 49561, 15 pp.

#### Cited by

1. A new family of convex weakly compact valued random variables in Banach space and applications to laws of large numbers vol.82, pp.1, 2012, https://doi.org/10.1016/j.spl.2011.08.012