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On the Law of The Iterated Logarithm for Rapidly Increasing Subsequences

✍ Scribed by Rainer Schwabe; Allan Gut

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
490 KB
Volume
178
Category
Article
ISSN
0025-584X

No coin nor oath required. For personal study only.

✦ Synopsis

The usual law of the iterated logarithm states that the partial sums Sn of independent and identically distributed random variables can be normalized by the sequence an = d -, such that limsup,,, &/a, = t/z a. 9.. As has been pointed out by GUT (1986) the law fails if one considers the limsup along subsequences which increase faster than exponentially. In particular, for very rapidly increasing subsequences {nk, k 2 1) one has limsupk-oo S,, /ank = 0 a. 9.. In these cases the normalizing constants ank have to be replaced by J;6;i.pk to obtain a non-trivial limiting behaviour: limsupk4ao S n k / d a = 0 a. 8. . We will present an intelligible argument for this structural change and apply it to related results.

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