Search

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: 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]

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]

Abstract: Let k and m be two fixed positive integers with and . Let which is the th smallest value of k real numbers . We call this f the th order function of k real numbers . Let be a nonconstant random variable with cumulative distribution . Let be the unique solution in (0, 1) to the equation where , which is the (cumulative) Beta distribution function with parameters m and . Write and . Define the hierarchical sequence of random variables by with being independent random variables identically distributed as . In this note it is shown that and a.s. and a.s. It follows that a.s. if and only if . This result generalizes and improves Propositions 4.3 and 4.4 of Li and Rogers [1999. Asymptotic behavior for iterated functions of random variables. Ann. Appl. Probab. 9, 1175–1201]. [Copyright &y& Elsevier]