DOI QR코드

DOI QR Code

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

  • Sung, Soo-Hak (Department of Applied Mathematics, Pai Chai University) ;
  • Hu, Tien-Chung (Department of Mathematics, National Tsing Hua University) ;
  • Volodin, Andrei I. (Department of Mathematics and Statistics, University of Regina)
  • Published : 2006.08.01

Abstract

Sung et al. [13] obtained a WLLN (weak law of large numbers) for the array $\{X_{{ni},\;u_n{\leq}i{\leq}v_n,\;n{\leq}1\}$ of random variables under a Cesaro type condition, where $\{u_n{\geq}-{\infty},\;n{\geq}1\}$ and $\{v_n{\leq}+{\infty},\;n{\geq}1\}$ 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

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 https://doi.org/10.1016/S0167-7152(97)85593-9
  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 https://doi.org/10.1016/0167-7152(92)90209-N
  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 https://doi.org/10.1214/aop/1176996029
  5. 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 https://doi.org/10.1016/0167-7152(94)00047-C
  6. 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 https://doi.org/10.1016/S0167-7152(99)00103-0
  7. 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
  8. 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
  9. 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
  10. F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374 https://doi.org/10.2140/pjm.1961.11.347
  11. S. H. Sung, Weak law of large numbers for arrays, Statist. Probab. Lett. 38 (1998), no. 2, 101-105 https://doi.org/10.1016/S0167-7152(97)00159-4
  12. 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 https://doi.org/10.1016/j.spl.2004.12.019
  13. 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