  For a non-uniform lattice in SL(2,R), we consider excursions in cusp
neighborhoods of a random geodesic on the corresponding finite area hyperbolic
surface or orbifold. We prove a strong law for a certain partial sum involving
these excursions. This generalizes a theorem of Diamond and Vaaler for
continued fractions. In the Teichmuller setting, we consider invariant measures
for the SL(2,R) action on the moduli spaces of quadratic differentials. By the
work of Eskin and Mirzakhani, these measures are supported on affine invariant
submanifolds of a stratum of quadratic differentials. For a Teichmuller
geodesic random with respect to a SL(2,R)-invariant measure, we study its
excursions in thin parts of the associated affine invariant submanifold. Under
a regularity hypothesis for the invariant measure, we prove similar strong laws
for certain partial sums involving these excursions. The limits in these laws
are related to the volume asymptotic of the thin parts. By Siegel-Veech theory,
these are given by various Siegel-Veech constants. As a direct consequence, we
show that the word metric grows faster than T log T along Teichmuller geodesics
random with respect to the Masur-Veech measure.
