which upon the time integration generates the following stochastic process $$E_q(t/\tau) = \int_0^t d{t^\prime} B \epsilon_q(t/\tau),$$ with the cumulant-generating function and the probability density, respectively, $$\begin{aligned}
{\label{eq:exp}}
    \kappa(\tilde{E}_q) &=& \ln[q + (1-q)/(1-B \tilde{E}_q)] t/\tau,\nonumber\\
    p(\tilde{E}_q) &=& q^{t/\tau} \delta(E_q/B)\nonumber\\
        &+& \sum_{i=0}^{\infty} \binom{t/\tau}{i} q^{t/\tau-i}(1-q)^{i} p_\Gamma(E_q;i,B)\text{,}
        \nonumber\\\end{aligned}$$ where $p_\Gamma(\cdot;i,B)$ is a probability density of the Gamma distribution with the shape parameter $i$ and the scale $B$.

The stochastic noise $\epsilon_q(t/\tau)$ reduces to simple exponential noise when $q = 0$, as it does under the conditions of random walk problem assumed in [Ref. [@PRE2016II]]{}. The general case of $q\ne0$ introduces one more parameter into [Eq. (\[eq:lgv2\])]{}, which is unpractical within the statistical precision of our model; cf. Secs. \[sec:mic\] and \[sec:mes\].

A shot noise approximation of the athermal noise [Eq. (\[eq:exp\])]{} can be derived in two ways. First, we may alter slightly the interpretation of a continuous limit for the random walk. Above the walker attempts $dn = dt / \tau$ steps in an infinitesimal time $dt$ [*deterministically*]{}. Instead, we could require, that the walker attempts a step with a probability proportional to a time interval $dt$, so small that it can make at most one displacement forward with the probability $dt/{\tau^\prime}$, or to 