Gaussian and simple exponential noise, to obtain a process ${P^\prime}(t)$ $$\kappa({\tilde{P}^\prime}) = \frac{a_2 B {\tilde{P}^\prime}t}{(1-B \tilde{P}) {\tau^\prime}},$$ where again the dimensionless constant $a_2$ can be absorbed into $\tau = {\tau^\prime}/a_2$ to yield [Eq. (\[eq:pois\])]{}.

Finally, we would like to remark, that the Central Limit Theorem ensures that the above approximations hold asymptotically. Indeed, all the random walk problems considered here tend to a Gaussian probability distribution for $n\to\infty$. Our Padé approximants have the same asymptotic limit, since they are accurate up to the second-order statistics.

Above we demonstrated, how the Padé approximation can be applied to formulate a random noise with presupposed statistical properties. In principle, however, other forms of approximants could be used in exactly the same way. For example, [Ref. [@DanchenkoChunaev2011]]{} suggests a general approach, which allows to generate various mathematical series, which fit the Taylor expansion of the approximated function. In particular, one of these series can be generated from the simple fraction $(1-x)^{-1}$ [@DanchenkoChunaev2011]. Under certain conditions, this yields a hyperexponential approximant for the moment-generating function $\mu(\tilde\xi)$ [@Hyperexp]. In other words, the Central Limit Theorem, as used in this section, and the approach of [Ref. [@DanchenkoChunaev2011]]{} give rise to a multitude of stochastic processes, which can 