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Motivated by an investigation of finding necessary and sufficient conditions for the Kolmogorov strong law of large numbers for the general Gini's mean difference, new versions of the classical Lévy, Ottaviani, and Hoffmann-J\ooTrgensen inequalities are obtained for a sequence of Banach space valued random variables. No geometric conditions are imposed on the Banach space. An application to the general Gini's mean difference in a Banach space setting is presented. [ABSTRACT FROM PUBLISHER]

Let {Yn, n ≥ 1} be a sequence of i.i.d. random variables and let l and L denote the essential infimum of Y1 and the essential supremum of Y1, respectively. The set  of almost sure limit points of [image omitted] where b > 1 is investigated. The new findings are for the case where Y1 is unbounded and are as follows: (i) If  ∩  ≠ ∅, then  = [l, L]; (ii) If l ∈ , then either  = {∞} or  = [l, ∞]; (iii) If l = - ∞ and L = ∞, then either  = {∞},  = {- ∞},  = [- ∞, ∞], or  = {- ∞, ∞}. Illustrative examples are referenced or provided showing that each of the various alternatives can hold. The current work is a continuation of the investigations of Li et al. [3, 4] wherein the set  is identified, respectively, for bounded Y1 as the spectrum of the distribution function of [image omitted] and for unbounded Y1 with (log (max {|Y1|, e})) < ∞ as [l, L]. [ABSTRACT FROM AUTHOR]

For a sequence of i.i.d. mean 0 random variables {X, Xn; n ≥ 1} with partial sums [image omitted], necessary and/or sufficient conditions are provided for {X, Xn; n ≥ 1} to enjoy iterated logarithm type behavior of the form [image omitted] almost surely where h(·) is a positive, nondecreasing function that is slowly varying at infinity. New results are {obtained for the cases X2 < ∞ and X2 = ∞. The proofs rely heavily on recent work of Einmahl and Li (Annals of Probability (2005) 33:1601-1624) and new versions of those results are obtained under conditions couched in terms of an “integral test” involving whether [image omitted] is finite or infinite where Lx = loge(e ∨ x) for x ≥ 0. Corollaries are presented for particular choices of h(·). [ABSTRACT FROM AUTHOR]