Exponential mixing of all orders and CLT for generic birational maps of ${\mathbb{P}}^k$

For Hénon maps, Bianchi and Dinh recently proved the exponential mixing of all orders for the measure of maximal entropy and, as a consequence of the recent work of Björklund and Gorodnik, the CLT for Hölder observables. We extend their results to generic birational maps of ${\mathbb{P}}^k$. Because of the indeterminacy set, Hölder maps are not stable under iteration, so we need to work with a suitable space of test functions.