**Appendix B**

In our model, we can consider waiting times between extreme events. We define an extreme event as an event occurring on average every *N* steps. The autocovariance of waiting times between *n* extreme events *COV*(*n*) can be written as

$$COV(n) = \sum\_{W=n-1}^{\infty} \sum\_{K\_1=1}^{\infty} \sum\_{K\_2=1}^{\infty} \sum\_{i=1}^{K\_1} \sum\_{j=1}^{K\_2} NB(W; n-1)NB(K\_1; 1)NB(K\_2; 1)cov(j + i + \mathcal{W} - 1),\tag{A17}$$

where *cov*(·) is defined by Equation (5) and *NB*(*k*, *n*) is a negative binomial distribution with *k* trials, given *n* successes and the probability of success 1*N*:

$$NB(k,n) = \binom{k-1}{n-1} \left(\frac{1}{\langle N \rangle}\right)^n \left(1 - \frac{1}{\langle N \rangle}\right)^{k-n}.\tag{A18}$$

The simpler form of this autocovariance can be derived thus:

$$\mathcal{COV}(n) = \left(\frac{1}{\langle N \rangle}\right)^{n-1} \sum\_{x=0}^{\infty} \left(1 - \frac{1}{\langle N \rangle}\right)^{x} \cot(x+n) \binom{x+n}{n}.\tag{A19}$$

Next, we calculate its *Z*-transform

$$
\widehat{\mathcal{C}O}\widehat{\mathcal{V}}(z) = \langle N \rangle \,\widehat{c\sigma}\widehat{\upsilon} \left( \frac{1}{\frac{\langle N \rangle - 1}{\langle N \rangle} + \frac{1/\langle N \rangle}{z}} \right), \tag{A20}
$$

where

$$\widehat{co}\widehat{w}(z) = \frac{1}{\zeta(\rho - 1)} \frac{z}{(z - 1)^2} \left[ (z - 1)\zeta(\rho - 1) - \zeta(\rho) + Li\_{\rho, z^{-1}} \right] \tag{A21}$$

and *Liρ*,*z*<sup>−</sup><sup>1</sup> is a polylogarithm function. Setting *z* = exp(*s*), we can show that the most important power-law term is *sρ*−3, which corresponds to a power-law decay *COV*(*n*) ∼ *n* −(*ρ*−<sup>2</sup>), analogically to Equation (8).
