prove theorem C below.

Almost Sure Invariance Principle {#bibi}
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Let $\psi \in L^2(\mu)$ be a $\mu$-centered observable and $S_n(\psi):= \sum_{j=0}^{n-1} \psi \circ f^j$. We say that $\psi$ satisfies the Almost Sure Invariance Principle (ASIP) if there exist, on an extended probability space, a sequence of random variables $(\SS_n)_{n \geq 0}$ together with a Brownian motion $\WW$ such that for some $\gamma > 0$:

1.  $\SS_n =  \WW(n) + o(n^{1/2 -
      \gamma})$ almost everywhere,

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

We shall denote ${\sigma}$-ASIP to specify the variance of Brownian motion.

[C:]{} For every $\mu$-centered observable $\psi \in \UU$, we have:

1.  ${\sigma}:= \lim_{n \to \infty} {1 \over \sqrt n} \norm{S_n(\psi)}_2$ exists, and ${\sigma}^2 = \int_{{\mathbb {P}}^k} \psi^2 \, d\mu + 2\sum_{j \geq 1} \int_{{\mathbb {P}}^k}
     \psi \cdot \psi \circ f^j \, d\mu$.

2.  If ${\sigma}= 0$, then $\psi = u - u \circ f$ holds $\mu$-a.e. for some $u \in L^2(\mu)$.

3.  If ${\sigma}> 0$, then $\psi$ satisfies the ${\sigma}$-ASIP.

The ASIP implies classical limit theorems related to Brownian motion: the Central Limit Theorem (CLT), the Law of Iterated Logarithm, Kolmogorov integral tests (see [@De], [@PS]). The ASIP also implies the almost 