random process $(\xi_n)_{n \geq 0}$ defined by $\xi_n({{\tilde \alpha}}) = {\alpha}_n$.

[(Philipp-Stout)]{} Let $\chi$ be a $\nu$-centered observable on $\Sigma$ satisfying:

1.  $\chi \in L^{2+\delta} (\nu)$ for some $\delta > 0$,

2.  $\norm{  \chi - {\mathbb {E}}(\chi \vert \CC_n)  }_{2+\delta} \leq c \, \beta ^n$ for some $c > 0$ and $\beta < 1$.

Then the sequence ${1 \over \sqrt n} \norm{S_n(\chi)}_2$ has a limit ${\sigma}$. If ${\sigma}> 0$, then $\chi$ satisfies the ${\sigma}$-ASIP.

Let us compare that result with Ibragimov’s theorem (see [@I], Theorem 2.1), which only requires moments of order $2$ and a summability condition:

[(Ibragimov)]{} Let $\chi$ be a $\nu$-centered observable on $\Sigma$ satisfying: $$\sum_{n \geq 0}
  \norm{  \chi - {\mathbb {E}}(\chi \vert \CC_n)  }_2 < \infty .$$ Then the sequence ${1 \over
    \sqrt n} \norm{S_n(\chi)}_2$ has a limit ${\sigma}$. If ${\sigma}> 0$, then $\chi$ satisfies the ${\sigma}$-CLT.

A quantified version of Briend-Duval theorem {#BD}
============================================

This section is devoted to the proof of theorem \[good\] (see subsection \[cons\]). That result will be crucial to establish theorem A.

Briend-Duval theorem {#cons}
--------------------

We recall that $\VV_l = \cup_{i = 1}^ l f^i(\CC)$, $\VV = \cup_{i = 0}^\infty
f^i(\CC)$ and that $d_t = d^k$ is the topological degree 