Probability measures related to geodesics in the space of K\"ahler metrics

We associate certain probability measures on $\mathbb{R}$ 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.