Refinements of Kolmogorov's law of the iterated logarithm

R. J. Tomkins

Statistics & Probability Letters, 1992, vol. 14, issue 4, 321-325

Abstract: Let \s{Sn\s} be the partial sums of a sequence \s{Xn\s} of centred random variables. Suppose s2n = ES2n, t2n = 2 log log s2n and sn --> [infinity]. It is shown that the law of the iterated logarithm (LIL) holds when tnXn/sn --> 0 almost surely and tn\vbXn\vb/sn [less-than-or-equals, slant] Y for all n [greater-or-equal, slanted] 1 and some L2-integrable Y, even though it may fail if only one of the conditions holds. Moreover, when tnXn/sn --> 0 a.s. and EX2n/s2n --> 0, the Central Limit Theorem implies the LIL, but the converse is not always true.

Keywords: Sums; of; independent; random; variables; law; of; the; iterated; logarithm; Central; Limit; Theorem (search for similar items in EconPapers)
Date: 1992
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