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Let $X_1,X_2,\dots$ be independent and identically distributed real random variables with $\mathbb{E}{X}_1=0$ and $\mathrm{Var}(X_1)={\sigma }^2\in (0,\infty )$. Write $S_n={\sum }_{j=1}^n{X}_j$. The iid law of the iterated logarithm statesFinite variance is sufficient in this iid theorem; the lower conclusion follows by applying the upper theorem to $-X_j$. The iterated logarithm is the slowly growing factor $\log \log n$.^[ref]

For a simple symmetric random walk, ${\sigma }^2=1$. Its scaled partial sums exceed every fixed multiple of $\sqrt{n}$ along some times almost surely, even though the central limit theorem gives a tight normal limit for $S_n/\sqrt{n}$ at each growing deterministic time. The iterated-logarithm law concerns the extremes encountered along one infinite realization, rather than only the distribution at one time.

It also implies $S_n/n\to 0$, since $\sqrt{\log \log n/n}\to 0$, and therefore refines the strong law of large numbers. For the sample mean, the almost-sure envelope is $\sigma \sqrt{2\log \log n/n}$. The upper law of the iterated logarithm and Brownian upper law of the iterated logarithm articles isolate one-sided parts of the argument; proving a summable upper-tail bound alone does not establish the matching lower excursions.