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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]

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]