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We consider the Hamiltonian system consisting of a Klein-Gordon vector fieldand a particle in $\R^3$. The initial date of the system is a random functionwith a finite mean density of energy which also satisfies a Rosenblatt- orIbragimov-type mixing condition. Moreover, initial correlation functions aretranslation-invariant. We study the distribution $\mu_t$ of the solution attime $t\in\R$. The main result is the convergence of $\mu_t$ to a Gaussianmeasure as $t\to\infty$, where $\mu_\infty$ is translation-invariant.

Convergence to equilibrium distribution. The Klein-Gordon equation coupled to a particle