Summary
For a sequence of independent random variables {X n} with zero means and finite variances, define \(S_n = \sum\limits_{j = 1}^n {X_j , s_n^2 = E(S_n^2 )}\) and t 2n =2 loglog s 2n ; assume s n→∞. Kolmogorov's law of the iterated logarithm asserts that lim sup S n/(sntn)=1 a.s. if t n¦Xn¦≦ɛ nsn for some real sequence n→∞ ɛn→0. This paper will show that, under the weaker condition t nXn/sn→0 a.s., the a.s. limiting value of lim sup S n(sntn) depends on the limiting behaviour of the modified Lindeberg functions
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This research was supported by grant A7588 from the Natural Sciences and Engineering Research Council of Canada.
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Tomkins, R.J. Lindeberg functions and the law of the iterated logarithm. Z. Wahrscheinlichkeitstheorie verw Gebiete 65, 135–143 (1983). https://doi.org/10.1007/BF00535000
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DOI: https://doi.org/10.1007/BF00535000