= Law of the iterated logarithm
{wiki}
Let $X_1,X_2,\ldots$ be <independent and identically distributed> real <random variables> with $\mathbb EX_1=0$ and $\operatorname{Var}(X_1)=\sigma^2\in(0,\infty)$. Write $S_n=\sum_{j=1}^nX_j$. The <independent and identically distributed>[iid] <law of the iterated logarithm> states
$$
\limsup_{n\to\infty}\frac{S_n}{\sqrt{2\sigma^2n\log\log n}}=1,
\qquad
\liminf_{n\to\infty}\frac{S_n}{\sqrt{2\sigma^2n\log\log n}}=-1
\quad\text{almost surely}.
$$
Finite <variance> is sufficient in this <independent and identically distributed>[iid] theorem; the lower conclusion follows by applying the upper theorem to $-X_j$. The iterated <logarithm> is the slowly growing factor $\log\log n$.<Probability: Theory and Examples (fifth edition) by Rick Durrett>[Section 8.5, Theorem 8.5.2, printed p. 418; negative excursions follow by sign reversal]{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 $S_n/n\to0$, since $\sqrt{\log\log n/n}\to0$, 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.
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