Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE CONVERGENCE OF SERIES OF MARTINGALE DIFFERENCES WITH MULTIDIMENSIONAL INDICES

  • SON, TA CONG (Faculty of Mathematics National University of Hanoi) ;
  • THANG, DANG HUNG (Faculty of Mathematics National University of Hanoi)
  • Received : 2014.11.28
  • Published : 2015.09.01
Download PDF
⟨ Previous Next ⟩

Abstract

Let {Xn; $n{\succeq}1$} be a field of martingale differences taking values in a p-uniformly smooth Banach space. The paper provides conditions under which the series ${\sum}_{i{\preceq}n}\;Xi$ converges almost surely and the tail series {$Tn={\sum}_{i{\gg}n}\;X_i;n{\succeq}1$} satisfies $sup_{k{\succeq}n}{\parallel}T_k{\parallel}=\mathcal{O}p(b_n)$ and ${\frac{sup_{k{\succeq}n}{\parallel}T_k{\parallel}}{B_n}}{\rightarrow\limits^p}0$ for given fields of positive numbers {bn} and {Bn}. This result generalizes results of A. Rosalsky, J. Rosenblatt [7], [8] and S. H. Sung, A. I. Volodin [11].

Keywords

References

  1. Y. S. Chow and H. Teicher, Probability Theory: Independence, Interchangeability, Martingales, 2nd ed., Springer-Verlag, New York, 1988.
  2. J. I. Hong and J. Tsay, A strong law of large numbers for random elements in Banach spaces, Southeast Asian Bull. Math. 34 (2010), no. 2, 257-264.
  3. N. V. Huan and N. V. Quang, The Doob inequality and strong law of large numbers for multidimensional arrays in general Banach spaces, Kybernetika (Prague) 48 (2012), no. 2, 254-267.
  4. E. Nam and A. Rosalsky, On the rate of convergence of series of random variables, Teor. Imovir. Mat. Stat. 52 (1995), 120-131 (in Ukrainian)
  5. English translation in: Theory Probab. Math. Statist. 52 (1996), 129-140.
  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, In: Probability and analysis (Varenna, 1985), 167-241, Lecture Notes in Math., 1206, Springer, Berlin, 1986.
  8. A. Rosalsky and J. Rosenblatt, On the rate of convergence of series of Banach space valued random elements, Nonlinear Anal. 30 (1997), no. 7, 4237-4248. https://doi.org/10.1016/S0362-546X(96)00217-9
  9. A. Rosalsky and J. Rosenblatt, On convergence of series of random variables with applications to martingale convergence and to convergence of series with orthogonal summands, Stoch. Anal. Appl. 16 (1998), no. 3, 553-566. https://doi.org/10.1080/07362999808809548
  10. A. Rosalsky and A. I. Volodin, On convergence of series of random elements via maximal moment relations with applications to martingale convergence and to convergence of series with p-orthogonal summands, Georgian Math. J. 8 (2001), no. 2, 377-388
  11. Correction, Georgian Math. J. 10 (2003), no. 4, 799-802.
  12. F. S. Scalora, Abstract martingale convergence theorems, Pacific J. Math. 11 (1961), 347-374. https://doi.org/10.2140/pjm.1961.11.347
  13. S. H. Sung and A. I. Volodin, On convergence of series of independent random variables, Bull. Korean Math. Soc. 38 (2001), no. 4, 763-772.

Cited by

  1. On convergence of moving average series of martingale differences fields taking values in Banach spaces pp.1532-415X, 2017, https://doi.org/10.1080/03610926.2017.1397172