An extension of Feller’s strong law of large numbers

This paper presents a general result that allows for establishing a link between the Kolmogorov–Marcinkiewicz– Zygmund strong law of large numbers and Feller’s strong law of large numbers in a Banach space setting. Let { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed Banach space valued random variables and set S n = ∑ i = 1 n X i , n ≥ 1 . Let { a n ; n ≥ 1 } and { b n ; n ≥ 1 } be increasing sequences of positive real numbers such that lim n → ∞ a n = ∞ and b n ∕ a n ; n ≥ 1 is a nondecreasing sequence. We show that S n − n E X I { ‖ X ‖ ≤ b n } b n → 0 almost surely for every Banach space valued random variable X with ∑ n = 1 ∞ P ( ‖ X ‖ > b n ) ∞ if S n ∕ a n → 0 almost surely for every symmetric Banach space valued random variable X with ∑ n = 1 ∞ P ( ‖ X ‖ > a n ) ∞ . To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables.