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http://dx.doi.org/10.4134/JKMS.2015.52.5.1023

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)

Publication Information

Journal of the Korean Mathematical Society / v.52, no.5, 2015 , pp. 1023-1036 More about this Journal

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

p-uniformly smooth Banach spaces; field of martingale differences; convergent of series of random field; tail series of random field;

Citations & Related Records

Reference

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