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

Let 0 < p < 2 and θ > 0. Let { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed B -valued random variables and set S n = ∑ i = 1 n X i , 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 sup n → ∞ log P ‖ S n ‖ > s n 1 / p (log n) θ = − p − θ ζ ¯ (p , θ) , lim inf n → ∞ log P ‖ S n ‖ > s n 1 / p (log n) θ = − p − θ ζ ̲ (p , θ) , where ζ ¯ (p , θ) = − lim sup t → ∞ log e p t P (log ‖ X ‖ > t) t θ and ζ ̲ (p , θ) = − lim inf t → ∞ log e p t P (log ‖ X ‖ > t) t θ . 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. [ABSTRACT FROM AUTHOR]