Rate of convergence in the multidimensional central limit theorem for stationary processes. Application to the Knudsen gas and to the Sinai billiard

We show how Rio’s method [Probab. Theory Related Fields 104 (1996) 255–282] can be adapted to establish a rate of convergence in $\frac{1}{\sqrt{n}}$ in the multidimensional central limit theorem for some stationary processes in the sense of the Kantorovich metric. We give two applications of this general result: in the case of the Knudsen gas and in the case of the Sinai billiard.