set $i = j = 1$, which is accurate at least up to the second-order statistics, for $i+j=2$, and has a general form [@BakerEssentials Chapter 1] $${\label{eq:MuPade1}}
    \mu_{[1/1]}(\tilde\xi) = \frac{a_{1,1} + a_{1,2} \tilde\xi}{a_{2,1} + a_{2,2} \tilde\xi},$$ where $a_{1,1}$, $a_{1,2}$, etc. are constants. The normalization of the probability distribution, though, requires that $\mu_{[1/1]}(0)=1$ and, thus, $a_{2,1} = a_{1,1}$. With $b_1 = a_{1,2}/a_{1,1}$ and $b_2 = a_{2,1}/a_{1,1}$, [Eq. (\[eq:MuPade1\])]{} becomes $${\label{eq:MuPade2}}
    \mu_{[1/1]}(\tilde\xi) = \frac{1 + b_1 \tilde\xi}{1 + b_2 \tilde\xi}
        = \frac{b_1}{b_2} + \frac{1-b_1/b_2}{1+b_2\tilde\xi}
        = q + \frac{1-q}{1-b\tilde\xi}.$$ where in the last equality we used another substitution $q=b_1/b_2$ and $B = -b_2$. The probability density, which corresponds to a moment-generating function $\mu_{[1/1]}(\tilde\xi)$, is $$p_{[1/1]}(\xi) = q \delta(\xi) + (1-q) H[\operatorname{sign}(B) \xi] \exp(-\xi/B)/B,$$ where $\delta(\cdot)$ and $H(\cdot)$ are, respectively, the Dirac delta and Heaviside step functions. The above equation is a valid probability density function $p_{[1/1]}(\xi)>0$ only for $0 \le q \le 1$. Furthermore, our assumptions $\kappa_1(\xi)>0$ and $\kappa_3(\xi)>0$ impose another restriction $B>0$. The continuous limit of the described approximation can be obtained by requiring, that the walker makes steps with a rate $\tau=dt/dn$, just like we did earlier in the derivation of $\omega(t)$. The result is a defintion of random noise $\epsilon_q(t/\tau)$, 