The distance between the Kac process and the Wiener process with applications to generalized telegraph equations (Q910805)

The author presents a local central limit theorem and establishes an expansion of length two for the Kac process \(Y_{\alpha}(t)\) describing the position of a particle at time t after collisions. The rate of convergence is obtained for the distance of total variation for the distributions of \(t^{-1/2}Y_{\alpha}(t)\) and the Wiener process at time t. The results are applied to the probabilistic solutions of abstract telegraph and heat equations. In particular, the rate of convergence is established for a singular perturbation problem of the abstract heat equation.