• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1023-1036
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• 2015

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].

• Journal of the Korean Mathematical Society
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• v.32 no.3
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• pp.427-446
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• 1995

Let S be a set and B be a $\sigma$-field on S. We consider $(\Omega = S^Z, T = B^z, P)$ as the basic probability space. We denote by T the left shift on $\Omega$. We assume that P is invariant under T, i.e., $PT^{-1} = P$, and that T is ergodic. We denote by $X = \cdots, X_-1, X_0, X_1, \cdots$ the coordinate maps on $\Omega$. From our assumptions it follows that ${X_i}_{i \in Z}$ is a stationary and ergodic process.