Editing Law of the iterated logarithm (section)

==Statement==
Let {''Y''<sub>''n''</sub>} be independent, identically distributed [[random variables]] with zero means and unit variances. Let ''S''<sub>''n''</sub> = ''Y''<sub>1</sub> + ... + ''Y''<sub>''n''</sub>. Then
: <math>
    \limsup_{n \to \infty} \frac{|S_n|}{\sqrt{2n \log\log n}} = 1 \quad \text{a.s.},
  </math>
where "log" is the [[natural logarithm]], "lim sup" denotes the [[limit superior]], and "a.s." stands for "[[almost surely]]".<ref>[[Leo Breiman]]. ''Probability''. Original edition published by Addison-Wesley, 1968; reprinted by Society for Industrial and Applied Mathematics, 1992. (See Sections 3.9, 12.9, and 12.10; Theorem 3.52 specifically.)</ref><ref>R. Durrett. ''Probability: Theory and Examples''. Fourth edition published by Cambridge University Press in 2010. (See Theorem 8.8.3.)</ref>

Another statement given by [[Andrey Kolmogorov|A. N. Kolmogorov]] in 1929<ref name="Kolmogorov1929" /> is as follows.

Let <math> \{ Y_n \} </math> be independent [[random variables]] with zero means and finite variances. Let <math> S_n = Y_1 + \dots + Y_n </math> and <math> B_n = \operatorname{Var}(Y_1) + \dots + \operatorname{Var}(Y_n) </math>. If <math> B_n \to \infty </math> and there exists a sequence of positive constants <math> \{ M_n \} </math> such that <math> |Y_n| \le M_n </math> a.s. and
: <math>
    M_n \;=\; o \left( \sqrt{\frac{B_n}{\log \log B_n}} \right),
  </math>
then we have
: <math>
    \limsup_{n \to \infty} \frac{|S_n|}{\sqrt{2 B_n \log\log B_n}} = 1 \quad \text{a.s.}
  </math>

Note that, the first statement covers the case of the standard normal distribution, but the second does not.