Journal of the
Korean Mathematical Society JKMS

Let $\{X_{\bf n};{\bf n\succeq 1}\}$ be a field of martingale differences taking values in a $p$-uniformly smooth Banach space. The paper provides conditions under which the series $\sum_{\bf i\preceq n}X_{\bf i}$ converges almost surely and the tail series $\{T_{\bf n}=\sum_{ \bf i \gg n}X_{\bf i}; {\bf n\succeq 1}\}$ satisfies $\sup_{\bf k\succeq n}\|T_{\bf k}\|=\mathcal{O}_{P}(b_{\bf n})$ and $\frac{\sup_{\bf k\succeq n}\|T_{\bf k}\|}{B_{\bf n}}\stackrel{P}{\to}0$ for given fields of positive numbers $\{b_{\bf n}\}$ and $\{B_{\bf n}\}$. This result generalizes results of A.~Rosalsky, J.~Rosenblatt \cite{h:2001}, \cite{i:2001} and S.~H.~Sung, A.~I.~Volodin \cite{g:2001}.

Keywords: $p$-uniformly smooth Banach spaces, field of martingale differences, convergent of series of random field, tail series of random field

MSC numbers: 60B11, 60B12, 60F15, 60G42, 60G60