  Consider the Klein-Gordon equation (KGE) in $\R^n$, $n\ge 2$, with constant
or variable coefficients. We study the distribution $\mu_t$ of the random
solution at time $t\in\R$. We assume that the initial probability measure
$\mu_0$ has zero mean, a translation-invariant covariance, and a finite mean
energy density. We also asume that $\mu_0$ satisfies a Rosenblatt- or
Ibragimov-Linnik-type mixing condition. The main result is the convergence of
$\mu_t$ to a Gaussian probability measure as $t\to\infty$ which gives a Central
Limit Theorem for the KGE. The proof for the case of constant coefficients is
based on an analysis of long time asymptotics of the solution in the Fourier
representation and Bernstein's `room-corridor' argument. The case of variable
coefficients is treated by using an `averaged' version of the scattering theory
for infinite energy solutions, based on Vainberg's results on local energy
decay.
