  Let $X_1, X_2, \dots$ be independent, identically distributed random
variables taking values from a compact metrizable group $G$. We prove that the
random walk $S_k=X_1 X_2 \cdots X_k$, $k=1,2,\dots$ equidistributes in any
given Borel subset of $G$ with probability $1$ if and only if $X_1$ is not
supported on any proper closed subgroup of $G$, and $S_k$ has an absolutely
continuous component for some $k \ge 1$. More generally, the sum $\sum_{k=1}^N
f(S_k)$, where $f:G \to \mathbb{R}$ is Borel measurable, is shown to satisfy
the strong law of large numbers and the law of the iterated logarithm. We also
prove the central limit theorem with remainder term for the same sum, and
construct an almost sure approximation of the process $\sum_{k \le t} f(S_k)$
by a Wiener process provided $S_k$ converges to the Haar measure in the total
variation metric.
