@article{oai:ynu.repo.nii.ac.jp:00006891,
 author = {Kruk, Lukasz and Zieba, Wieslaw},
 issue = {1},
 journal = {Yokohama Mathematical Journal = 横濱市立大學紀要. D部門, 数学},
 month = {},
 note = {application/pdf, In this paper we give a necessary and sufficient condition for a ＄L^{1}＄-bounded asymptotic martingale (amart) taking values in a Banach space to converge almost surely in norm: such an asymptotic martingale ＄(X_{n}, F_{n}, n￥geqq 1)＄ converges a.s. iff it is strongly tight, i.e. for every ＄￥epsilon>0＄ there exists a compact set ＄K_{￥epsilon}＄ such that ＄ (￥bigcap_{n=1}^{￥infty}[X_{n}￥in K_{￥epsilon}])>1-￥epsilon＄ . Moreover, we show that for realvalued martingales the well known theorem of Doob is, in some sense, the best possible-there exists a martingale ＄(X_{n}, n￥geqq 1)＄ such that ￥sup_{n}E|X_{n}|^{a}<￥infty＄ for every ＄a￥in(O, 1)＄ and it diverges a.s. (in fact, it does not even converge in law, although it is strongly tight).},
 pages = {61--72},
 title = {A CRITERION OF ALMOST SURE CONVERGENCE OF ASYMPTOTIC MARTINGALES IN A BANACH SPACE},
 volume = {43},
 year = {1995}
}