The law of iterated logarithms operates "in between" the
law of large numbers and the
central limit theorem. There are two versions of the law of large numbers —
the weak and
the strong — and they both state that the sums
Sn, scaled by
n−1, converge to zero, respectively
in probability and
almost surely: : \frac{S_n}{n} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{n} \ \xrightarrow{a.s.} 0, \qquad \text{as}\ \ n\to\infty. On the other hand, the central limit theorem states that the sums
Sn scaled by the factor
n−1/2 converge in distribution to a standard normal distribution. By
Kolmogorov's zero–one law, for any fixed
M, the probability that the event \limsup_n \frac{S_n}{\sqrt{n}} \geq M occurs is 0 or 1. Then : \Pr\left( \limsup_n \frac{S_n}{\sqrt{n}} \geq M \right) \geqslant \limsup_n \Pr\left( \frac{S_n}{\sqrt{n}} \geq M \right) = \Pr\left( \mathcal{N}(0, 1) \geq M \right) > 0 so :\limsup_n \frac{S_n}{\sqrt{n}}=\infty \qquad \text{with probability 1.} An identical argument shows that : \liminf_n \frac{S_n}{\sqrt{n}}=-\infty \qquad \text{with probability 1.} This implies that these quantities cannot converge almost surely. In fact, they cannot even converge in probability, which follows from the equality :\frac{S_{2n}}{\sqrt{2n}}-\frac{S_n}{\sqrt{n}} = \frac1{\sqrt2}\frac{S_{2n}-S_n}{\sqrt{n}} - \left (1-\frac1\sqrt2 \right )\frac{S_n}{\sqrt{n}} and the fact that the random variables :\frac{S_n}{\sqrt{n}}\quad \text{and} \quad \frac{S_{2n}-S_n}{\sqrt{n}} are independent and both converge in distribution to \mathcal{N}(0, 1). The
law of the iterated logarithm provides the scaling factor where the two limits become different: : \frac{S_n}{\sqrt{2n\log\log n}} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{\sqrt{2n\log\log n}} \ \stackrel{a.s.}{\nrightarrow}\ 0, \qquad \text{as}\ \ n\to\infty. Thus, although the absolute value of the quantity S_n/\sqrt{2n\log\log n} is less than any predefined
ε > 0 with probability approaching one, it will nevertheless almost surely be greater than
ε infinitely often; in fact, the quantity will be visiting the neighborhoods of any point in the interval (-1,1) almost surely. ==Generalizations and variants==