Almost Sure Invariance Principle (ASIP) if there exist on a probability space $(\tilde X , \tilde m)$ a sequence of random variables $(\SS_n)_{n \geq 0}$ and a Brownian motion $\WW$ such that:

1.  $\SS_n =  \WW(n) + o(n^{1/2 -
      \gamma})$ $\tilde m$-a.e. for some $\gamma > 0$,

2.  $(S_0(\psi),\ldots,S_n(\psi))$ and $(\SS_0,\ldots,\SS_n)$ have the same distribution for any $n \geq 0$.

We denote ${\sigma}$-ASIP to specify the variance of Brownian motion. The ${\sigma}$-ASIP implies the ${\sigma}$-Central Limit Theorem (${\sigma}$-CLT), meaning that: $$\forall t \in {\mathbb {R}}\ , \ \lim_{n \to \infty}  m \left( {S_n(\varphi)
  \over {\sigma}\sqrt n} \leq t  \right)  =  {1 \over \sqrt{2\pi}}
\int_{-\infty}^t e^{-u^2/2} \, du .$$

Suppose that ${\omega}: \Sigma \to {\mathbb {P}}^k$ is a coding map provided by theorem A. Since ${\omega}_* \nu = \mu$ and $f \circ {\omega}= {\omega}\circ s$, a $\mu$-centered observable $\psi \in L^{2}(\mu)$ satisfies the ${\sigma}$-ASIP if and only if the $\nu$-centered observable $\chi:= \psi \circ {\omega}\in
L^{2}(\nu)$ satisfies the ${\sigma}$-ASIP.

We shall use Philipp-Stout’s theorem ([@PS], Section 7) to prove the ASIP for $\chi
:= \psi \circ {\omega}$ on the Bernoulli space $(\Sigma,s,\nu)$. The version below comes from the original one by using the $s$-invariance of $\nu$ and the independence of the 