Law of The Iterated Logarithm - Discussion

Discussion

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 claim 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−½ converge in distribution to a standard normal distribution. Combined with Kolmogorov's zero-one law, this implies that these quantities converge neither in probability nor almost surely:

\frac{S_n}{\sqrt n} \ \stackrel{p}{\nrightarrow}\ \forall, \qquad \frac{S_n}{\sqrt n} \ \stackrel{a.s.}{\nrightarrow}\ \forall, \qquad \text{as}\ \ n\to\infty.

The law of the iterated logarithm provides the scaling factor where the two limits become different:

\frac{S_n}{\sqrt{n\log\log n}} \ \xrightarrow{p}\ 0, \qquad \frac{S_n}{\sqrt{n\log\log n}} \ \stackrel{a.s.}{\nrightarrow}\ 0, \qquad \text{as}\ \ n\to\infty.

Thus, although the quantity is less than any predefined ε > 0 with probability approaching one, that quantity will nevertheless be dropping out of that interval infinitely often, and in fact will be visiting the neighborhoods of any point in the interval (0,√2) almost surely.

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