to ensure that [Eq. (\[eq:gaussn\])]{} holds asymptotically in the limit of large $n$, at least for fluctuations of size $\sqrt{n}$. At last, the continuous limit of the described random walk is taken, by requiring that, in an infinitesimal time $dt$, the walker makes a number of steps $dn$ with a rate $\tau = dt/dn$, so that [Eq. (\[eq:gaussn\])]{} becomes $${\label{eq:gausst}}\kappa(\tilde{x}) \approx \frac{\kappa_2(\xi)  t \tilde{x}^2}{2\tau},$$ which is the cumulant-generating function for the process $\Omega(t)$ with $A^2 = \kappa_2(\xi)/\tau$.

Instead of simply reviewing exponential noise, however, we now derive a more general family of random processes, to which it pertains. To do this, we again consider a random walk problem $\xi\approx B\epsilon(t/\tau)dt$, similar to the one described above. Due to the physical assumptions discussed in [Ref. [@PRE2016II]]{}, this time we require $\kappa_3(\xi) > 0$ and that the probability density $p(\xi)$ vanishes for $\xi<0$, which implies $\kappa_1(\xi)\ge0$.

Earlier we used power series to approximate the cumulant-generating function for the symmetric random walk. This time we will use a Padé approximant [@BakerEssentials] for the moment-generating function $\mu(\tilde\xi)$.

Just like with power series, not all orders of the numerator and denominator, $i$ and $j$, respectively, in a Padé approximant $\mu_{[i/j]}(\tilde\xi)\approx\mu(\tilde\xi)$, are valid moment-generating functions. We 