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For an array {Vn,j,1 ≤ j ≤ kn,n ≥ 1} of random elements taking values in a real separable Rademacher type p (1 < p ≤ 2) Banach space and a sequence of positive constants {dn,n ≥ 1}, a theorem is established providing conditions under which the degenerate mean convergence result ... holds where Sn = ∑nj=1 Vn,j, n ≥ 1. An example is provided showing that the above degenerate mean convergence can fail if the Banach space is not of Rademacher type p where 1 < p 2. Moreover for a general sequence of random elements {Wn,n ≥ 1} which is not structurally of any specific form taking values in a real separable Banach space which is not assumed to be of Rademacher type p for any p ∈ (1,2], conditions are provided under which the degenerate mean convergence result E(g(||Wn||)) → 0 holds where g is a continuous strictly increasing function with g(0) = 0 and limx→∞ g(x)= ∞. [ABSTRACT FROM AUTHOR]

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Let 0 < p < 2 and 1 ≤ q < ∞. Let {Xn; n ≥ 1} be a sequence of independent copies of a real-valued random variable X and set Sn = X1 + · · ·+Xn, n ≥ 1. We say X satisfies the (p, q)-type strong law of large numbers (and write X ∈ SLLN(p, q)) if ... < ∞ almost surely. This paper is devoted to a characterization of X ∈ SLLN(p, q). By applying results obtained from the new versions of the classical Lévy, Ottaviani, and Hoffmann-Jørgensen (1974) inequalities proved by Li and Rosalsky (2013) and by using techniques developed by Hechner (2009) and Hechner and Heinkel (2010), we obtain sets of necessary and sufficient conditions for X ∈ SLLN(p, q) for the six cases: 1 ≤ q < p < 2, 1 < p = q < 2, 1 < p < 2 and q > p, q = p = 1, p = 1 < q, and 0 < p < 1 ≤ q. The necessary and sufficient conditions for X ∈ SLLN(p, 1) have been discovered by Li, Qi, and Rosalsky (2011). Versions of the above results in a Banach space setting are also given. [ABSTRACT FROM AUTHOR]

Letbe a sequence of independent and identically distributedB-valued random variables, and set. Let,andq>0, and putWe strengthen the convergence rate for the Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers in Banach space, by showing that, if and only ifand [ABSTRACT FROM PUBLISHER]

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]