ON THE CONVERGENCE OF SERIES OF MARTINGALE DIFFERENCES WITH MULTIDIMENSIONAL INDICES

Title & Authors
ON THE CONVERGENCE OF SERIES OF MARTINGALE DIFFERENCES WITH MULTIDIMENSIONAL INDICES
SON, TA CONG; THANG, DANG HUNG;

Abstract
Let {Xn; $\small{n{\succeq}1}$} be a field of martingale differences taking values in a p-uniformly smooth Banach space. The paper provides conditions under which the series $\small{{\sum}_{i{\preceq}n}\;Xi}$ converges almost surely and the tail series {$\small{Tn={\sum}_{i{\gg}n}\;X_i;n{\succeq}1}$} satisfies $\small{sup_{k{\succeq}n}{\parallel}T_k{\parallel}=\mathcal{O}p(b_n)}$ and $\small{{\frac{sup_{k{\succeq}n}{\parallel}T_k{\parallel}}{B_n}}{\rightarrow\limits^p}0}$ 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;
Language
English
Cited by
References
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