  For H\'enon 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 Bj\"orklund and Gorodnik, the CLT for H\"older observables.
  We extend their results to generic birational maps of $\mathbb{P}^k$. Because
of the indeterminacy set, H\"older maps are not stable under iteration, so we
need to work with a suitable space of test functions.
