Law of The Iterated Logarithm - Statement

Statement

Let {Yn} be independent, identically distributed random variables with means zero and unit variances. Let Sn = Y1 + … + Yn. Then

\limsup_{n \to \infty} \frac{S_n}{\sqrt{n \log\log n}} = \sqrt{2}, \qquad \text{a.s.},

where “log” is the natural logarithm, “lim sup” denotes the limit superior, and “a.s.” stands for “almost surely”.

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