  We study the fluctuations of self-intersection counts of random geodesic
segments of length $t$ on a compact, negatively curved surface in the limit of
large $t$. If the initial direction vector of the geodesic is chosen according
to the \emph{Liouville measure}, then it is not difficult to show that the
number $N (t)$ of self-intersections by time $t$ grows like $\kappa t^{2}$,
where $\kappa =\kappa_{M}$ is a positive constant depending on the surface $M$.
We show that (for a smooth modification of $N (t)$) the fluctuations are of
size $t$, and the limit distribution is a weak limit of Gaussian quadratic
forms. We also show that the fluctuations of \emph{localized} self-intersection
counts (that is, only self-intersections in a fixed subset of $M$ are counted)
are typically of size $t^{3/2}$, and the limit distribution is Gaussian.
