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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]

In this paper, a general large deviation result for partial sums of independent and identically distributed random variables with super-heavy tailed distribution is established. Our main result extends in particular the results of Stoica (2008) and Nakata (2022). The symmetrization technique, one of Lévy's inequalities, and two preliminary results on slowly and regularly varying functions are paramount in the proof of our main result. [ABSTRACT FROM AUTHOR]