2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/125744In a recent paper by the authors a general result characterizing two-sided LIL behavior for real valued random variables has been established. In this paper, we show that there are analogous results in the Banach space setting. One of our main new tools is an improved Fuk-Nagaev type inequality in Banach space which should be of independent interest.The Fuk-Nagaev inequality and he upper/ lower bound in Theorem 5 have been improvedProbability60B12, 60F15text