Abstract: Let {X_n;n≥1}be a sequence of m-dependent Banach space valued random variables with mean zeros and finite variances, by using the weak convergence theorem of m-dependent Banach space valued random variables, we studied the precise asymptotics of the complete convergence and the law of the iterated logarithm for the sequence {X_n;n≥1}. We extended the results of the precise asymptotics of the complete convergence and the law of the iterated logarithm for real valued i.i.d.random variables.

Key words: m-dependent random variable, complete convergence, precise asymptotics