statistical properties. In fact both, simple exponential noise and exponential shot noise, belong to two families of random processes, which can be obtained in this manner.

Let us begin with a review of the Gaussian white noise $\omega(t)$, which is defined as a time derivative of the Gaussian random process $\Omega(t)$ with the zero mean and the variance $A^2 t$, so that $$\Omega(t) = \int_0^{t} d{t^\prime} A \omega({t^\prime})\text{,}$$ where $A>0$ is a constant.

In the derivation of Gaussian white noise we consider a random walk, where the walker moves in a discrete series of steps from an initial position $x_0 = 0$. The length of each step $\xi\approx A \omega(t) dt$ is an independent random variable with a given probability density $p(\xi)$. A physically relevant property, which we want to incorporate in this model, is a spatial symmetry for the forward ($\xi>0$) and backward ($\xi<0$) displacements. Statistically this means $p(\xi) = p(-\xi)$. The walker’s position $x$ after $n$ steps is then given by a sum of $n$ independent realizations of $\xi$. Therefore, the moment-generating function of the random variable $x$ equals the moment-generating function of $\xi$ raised to the power $n$, and thus $$\begin{aligned}
    {\label{eq:mu}} \mu(\tilde{x}) &=& [\mu(\tilde\xi)]^n\vline_{\tilde\xi=\tilde{x}};\\
    {\label{eq:kappa}} \kappa(\tilde{x}) 