# Autocorrelated Inter-arrival Times of Extreme Events

I'm using a bunch of techniques and methods from Extreme Value Theory to analyze my data. I have a time series representing the number of events happening in a given day. The time series is unequally temporally spaced, so that I have multiple observations for some days, and zero observations for other days. However, no missing values: if I do not have any observations for a given day, it means that no events happened that day.

The number of events distribution shows a Pareto behavior. I do not attach the plots, but I verified it with qqplots, mean excess plots, Zipf plot, etc. Moreover, I found out that the moments greater than 1 do not exist, so that I should expect that my distribution is in the Maximum Domain of Attraction of the Fréchet distribution, with $\alpha \in [1,2)$.

To estimate $\alpha$ and investigate the dynamics generating the number of events, I used Peaks Over Threshold (POT). Such an approach has two main consequences:

1. the exceedances (when extreme events happen) are governed by a homogeneous Poisson process, so that the inter-arrival times between the exceedances are exponentially distributed;
2. the number of events exceeding the threshold follow a GPD (Generalized Pareto Distribution).

One important assumption is that the observations are IID.

I know well the phenomenon under investigation: no way that the number of events happening on a given day can affect the number of events happening in other days. I'm sure. It would make no sense. However, I check for autocorrelation using ACF plot. I can safely conclude that there is no autocorrelation in my time series. See the figure below. Next, in order to estimate the GPD parameters, I used shape function from evir (R package) to choose a proper threshold $L$. $L = 5$ looks like a suitable choice, since the value of $\xi$ is somehow stable and the $95\%$ confidence intervals are narrow.

Using $L = 5$ I obtained the following (nice) estimates:

• $2,430$ exceedances ($0.917\%$ of the total number of observations)
• $\xi = 0.84 \pm 0.0376$
• $\sigma = 5.08 \pm 0.1964$
• $\mu = 6.30 \pm 0.1134$

Morover, since $\alpha = \frac{1}{\xi}$, $\alpha = 1.19$ (a value consistent with my exploratory analysis). The Pareto tail fit is shown in the figure below. However, the inter-arrival times between exceedances are very far from being exponentially distributed. See figure below, where the concave departure indicates a Pareto behavior. Such a behavior is confirmed by the empirical cumulative distribution function (ECDF) and from the fact that all the moments are infinite. I can safely conclude that the exceedances' occurrences are not governed by a homogenoeus Poisson process, since their inter-arrival times are Pareto distributed.

Moreover, when I look at the ACF plot of the inter-arrival times, I see lag-1 to lag-5 dependences. See figure below. Indeed, if I look at the point process exceedances I see a sort of clustering of extreme events. See figure below. Such results do not depend on the choice of the threshold (I obtain the same results for $L \in \{1,2,10\}$).

Such a behavior does not make sense, I'm not analyzing earthquakes or similar extreme events that tend to cluster. I was expecting anything but extreme events clustering.

Am I missing something? Any ideas?

However, how can I model that kind of inter-arrival times (autocorrelated and Pareto-distributed with infinite moments)?