  We associate certain probability measures on $\R$ to geodesics in the space
$\H_L$ of positively curved metrics on a line bundle $L$, and to geodesics in
the finite dimensional symmetric space of hermitian norms on $H^0(X, kL)$. We
prove that the measures associated to the finite dimensional spaces converge
weakly to the measures related to geodesics in $\H_L$ as $k$ goes to infinity.
The convergence of second order moments implies a recent result of Chen and Sun
on geodesic distances in the respective spaces, while the convergence of first
order moments gives convergence of Donaldson's $Z$-functional to the Aubin-Yau
energy. We also include a result on approximation of infinite dimensional
geodesics by Bergman kernels which generalizes work of Phong and Sturm.
