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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: We prove a Kolmogorov–Feller weak law of large numbers for exchangeable sequences, under a second order hypothesis on the truncated mixands. [Copyright &y& Elsevier]