the equidistributed probability measure $\bar \nu$. We set $\Sigma:= \AA^{\mathbb {N}}$, $s : \Sigma \to \Sigma$ the left shift and $\nu:= \otimes_{n =
  0}^\infty \bar \nu$. We denote by ${{\tilde \alpha}}:= ({\alpha}_n)_{n \geq 0}$ the elements of $\Sigma$, by $\CC_n$ the set of cylinders of length $n+1$, and by $\pi_n : \Sigma \to \AA^{n+1}$ the projection $\pi_n
({{\tilde \alpha}}):= ({\alpha}_0 , \ldots , {\alpha}_n )$. For any ${{\tilde \alpha}}\in \Sigma$, we set $C_n({{\tilde \alpha}}): = \pi_n^{-1} ({\alpha}_0 , \ldots , {\alpha}_n
)$. We denote by ${\mathbb {E}}(\chi \vert \CC_n)$ the conditional expectation of $\chi \in L^2(\nu)$ with respect to $\CC_n$. If $\LL = \{ A_1 , \ldots
, A_p \} \subset \CC_n$, we set $\LL^*:= \cup_{1 \leq j \leq p} A_j$.

Almost Sure Invariance Principle {#consb}
--------------------------------

Let $(X,g,m)$ be either $(\Sigma,s,\nu)$ or $({\mathbb {P}}^k,f,\mu)$. For any observable $\varphi \in
L^2(m)$, we set $S_n(\varphi):= \sum_{j = 0}^{n-1} \varphi \circ
g^j$ and $R_j(\varphi):= \int_X \varphi \cdot \varphi \circ g^j \,
dm$. We say that $\varphi$ is $m$-centered if $\int_X \varphi \, dm =
0$ and that $\varphi$ is a cocycle if $\varphi = u - u \circ g$ $m$-a.e. for some $u \in L^2(m)$.\
An observable $\varphi$ on $(X,g,m)$ satisfies the 