A general logarithmic asymptotic behavior for partial sums of i.i.d. random variables

Let 0 < p < 2 and theta > 0. Let {X, X-n; n >= 1} be a sequence of independent and identically distributed B-valued random variables and set S-n = Sigma(n)(i=1) X-t, n >= 1. In this note, a general logarithmic asymptotic behavior for {S-n; n >= 1} is established. We show that if S-n/n(1/p) ->(P) 0, then, for all s > 0, {lim(n ->infinity) sup log P (parallel to S-n parallel to > sn(1/p)) /(log n)(theta) = -p(-theta)(zeta) over bar (p, theta), lim(n ->infinity) inf log P (parallel to S-n parallel to > sn(1/p)) /(log n)(theta) = -p(-theta)(zeta) under bar (p, theta), where (zeta) over bar (p, theta) = - lim(t ->infinity) sup log (e(pt)P(log parallel to X parallel to > t))/t(theta) and (zeta) under bar (p, theta) = -lim(t ->infinity) inf log (e(pt)P(log parallel to X parallel to > t))/t(theta). The main tools used to prove this result are the symmetrization technique, an auxiliary lemma for the maximum of i.i.d. random variables, a moment inequality, and an exponential inequality.