  In this paper we propose a model of random compositions of cylinder maps,
which in the simplified form is as follows: let $(\theta,r)\in \mathbb T\times
\mathbb R=\mathbb A$ and \[ f_{\pm 1}:
\left(\begin{array}{c}\theta\\r\end{array}\right) \longmapsto
\left(\begin{array}{c}\theta+r+\varepsilon u_{\pm 1}(\theta,r) \\ r+\varepsilon
v_{\pm 1}(\theta,r) \end{array}\right), \] where $u_\pm$ and $v_\pm$ are smooth
and $v_\pm$ are trigonometric polynomials in $\theta$ such that $\int
v_\pm(\theta,r)\,d\theta=0$ for each $r$. We study the random compositions \[
(\theta_n,r_n)=f_{\omega_{n-1}}\circ \dots \circ f_{\omega_0}(\theta_0,r_0), \]
where $\omega_k =\pm 1$ with equal probability. We show that under
non-degeneracy hypotheses and away from resonances for $n\sim \varepsilon^{-2}$
the distributions of $r_n-r_0$ weakly converge to a stochastic diffusion
process with explicitly computable drift and variance.
  In the case $u_\pm(\theta)=v_\pm(\theta)$ are trigonometric polynomials of
zero average we prove a vertical central limit theorem, namely, for $n\sim
\varepsilon^{-2}$ the distributions of $r_n-r_0$ weakly converge to the normal
distribution $\mathcal N(0,\sigma^2)$ with $\sigma^2=\frac14\int
(v_+(\theta)-v_-(\theta))^2\,d\theta$.}
  The considered random model up to higher order terms in $\varepsilon$ is
conjugate to a restrictions to a Normally Hyperbolic Invariant Lamination of
the generalized Arnold example. Combining the result of this paper with
[8,23,28] we show formation of stochastic diffusive behaviour for the
generalized Arnold example.
