Weak laws of large numbers for nonnegative independent random variables without finite means

Pingyan Chen and Soo Hak Sung

Communications in Statistics - Theory and Methods, 2019, vol. 48, issue 21, 5401-5413

Abstract: For a sequence of nonnegative independent random variables {Xn,n≥1} with P(Xn > x) ≈ ln(x)/x, where ln(x) > 0,n≥1, are slowly varying functions, we present some mild conditions on {ln(x)} under which ∑i=1nXi/bn→1 in probability, where {bn,n≥1} is a norming sequence of positive numbers with bn→∞ When the distributions of the random variables are specified more precisely, the norming sequence can be obtained more concretely. As special cases, we obtain the results of Adler (2012, 2017) and Nakata (2016).

Date: 2019
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DOI: 10.1080/03610926.2018.1513145

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