of the correlation function is by no means trivial. As demonstrated for example in ref.[@SPDan] the results obtained for the correlation function calculated with the trajectories of the dynamical system are, in many cases, not reliable and cannot be improved by going to longer propagation times. Therefore other methods for the calculation of the time correlations are needed. Here we use a numerical approach to eq.(\[corfct\]). It is based on the extraction of the monotony intervals for the $n$-th iterate. The endpoints of the monotony intervals are given as the preimages of the maximum $x_{\max}$. The integration is then performed for each monotony interval separately. This ensures an accurate although very CPU time consuming evaluation of the correlation function. The latter is related to the exponential growth ($2^n$) of the number of monotony intervals with increasing $n$.

We apply now the stochastic method described in the previous section to several examples to demonstrate its capability of providing dynamical systems with prescribed statistical properties. The four different cases we study here can be summarized with respect to their statistical properties as follows:

-   [exponentially decaying autocorrelation function and uniform density]{}

-   [exponentially decaying autocorrelation function and linear density]{}

-   [oscillatory decaying 