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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. [ABSTRACT FROM AUTHOR]

In this note, general symmetrization procedures for both the weak and strong laws of large numbers concerning the row sums from arrays of rowwise independent Banach space valued random variables are established. Necessary and sufficient conditions are provided for the weak and strong laws of large numbers to hold in terms of corresponding laws of large numbers for a symmetrized version of the array. Several corollaries of the main result are provided. [ABSTRACT FROM AUTHOR]

Let { X , X n ; n ≥ 1 } be a sequence of i.i.d. real-valued random variables, and let S n = ∑ i = 1 n X i , n ≥ 1 . Write log x = log e ( e ∨ x ) , x ≥ 0 . In this note we establish a limit theorem which is related to the classical Hartman–Wintner–Strassen law of the iterated logarithm and the classical Chover law of the iterated logarithm. That is, for 1 ≤ p < ∞ , we show that lim sup n → ∞ | S n n | ( log log log n ) − 1 = e p / 2 almost surely if and only if E X = 0 and inf { b > 0 : lim x → ∞ ( log log x ) 1 − b E ( X 2 I ( | X | ≤ x ) ) = 0 } = p . [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper by Chen (in press) the limit law of the iterated logarithm for the partial sums of i.i.d. real-valued random variables has been established. In this note we look at the corresponding problem in Banach space setting. Let be a real separable Banach space with topological dual . Let be a sequence of i.i.d. -valued random variables and set and . We show that provided that where and is the unit ball of . [Copyright &y& Elsevier]

Abstract: Let be a sequence of i.i.d. random variables, and set . For , if , and , we prove that thus extending a result by . For and , we establish conditions for the convergence of the related series and derive its precise asymptotic as . [Copyright &y& Elsevier]

Abstract: For a sequence of i.i.d. mean 0 random variables with weighted partial sums where is a Lipschitz function of order 1 with , necessary and sufficient conditions are provided for to enjoy iterated logarithm type behavior of the form almost surely where is a positive, nondecreasing function which is slowly varying at infinity. Some corollaries are presented for particular choices of . [Copyright &y& Elsevier]