  Let $f$ be an holomorphic endomorphism of $\mathbb{P}^k$ and $\mu$ be its
measure of maximal entropy. We prove an Almost Sure Invariance Principle for
the systems $(\mathbb{P}^k,f,\mu)$. Our class $\cal{U}$ of observables includes
the H\"older functions and unbounded ones which present analytic singularities.
The proof is based on a geometric construction of a Bernoulli coding map
$\omega: (\Sigma, s, \nu) \to (\mathbb{P}^k,f,\mu)$. We obtain the invariance
principle for an observable $\psi$ on $(\mathbb{P}^k,f,\mu)$ by applying
Philipp-Stout's theorem for $\chi = \psi \circ \omega$ on $(\Sigma, s, \nu)$.
The invariance principle implies the Central Limit Theorem as well as several
statistical properties for the class $\cal{U}$. As an application, we give a
\emph{direct} proof of the absolute continuity of the measure $\mu$ when it
satisfies Pesin's formula. This approach relies on the Central Limit Theorem
for the unbounded observable $\log \textsf{Jac} f \in \cal{U}$.
