A Multivariate Berry--Esseen theorem for time-dependent expanding dynamical systems

We adapt Stein's method to obtain Berry--Esseen type error bounds in the multivariate central limit theorem for non-stationary processes generated by time-dependent compositions of uniformly expanding dynamical systems. In a particular case of random dynamical systems with a strongly mixing base transformation, we derive an error estimate of order O (N^-1/2) in the quenched multivariate CLT, provided that the covariance matrix grows linearly with the number of summands N. The error in the normal approximation is estimated for the class of all convex sets.