Yokohama City University and Yokohama National University
Abstract
application/pdfIn 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).departmental bulletin pape
