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For a non-uniform lattice in SL(2,R), we consider excursions in cuspneighborhoods of a random geodesic on the corresponding finite area hyperbolicsurface or orbifold. We prove a strong law for a certain partial sum involvingthese excursions. This generalizes a theorem of Diamond and Vaaler forcontinued fractions. In the Teichmuller setting, we consider invariant measuresfor the SL(2,R) action on the moduli spaces of quadratic differentials. By thework of Eskin and Mirzakhani, these measures are supported on affine invariantsubmanifolds of a stratum of quadratic differentials. For a Teichmullergeodesic random with respect to a SL(2,R)-invariant measure, we study itsexcursions in thin parts of the associated affine invariant submanifold. Undera regularity hypothesis for the invariant measure, we prove similar strong lawsfor certain partial sums involving these excursions. The limits in these lawsare 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, weshow that the word metric grows faster than T log T along Teichmuller geodesicsrandom with respect to the Masur-Veech measure.

Partial sums of excursions along random geodesics and volume asymptotics for thin parts of moduli spaces of quadratic differentials