# Ordinal Patterns in Clusters of Subsequent Extremes of Regularly Varying Time Series

## TL;DR

This paper studies the structure of clusters of extreme events in time series, focusing on their size distribution and ordinal patterns, with methods applicable to real-world data like river discharge measurements.

## Contribution

It introduces non-parametric estimators for cluster size and ordinal patterns, proves their asymptotic properties, and demonstrates their effectiveness through simulations and real data.

## Key findings

- Limit distributions for cluster features are established.
- Estimators perform well in simulations.
- Application to river discharge data reveals meaningful patterns.

## Abstract

In this paper, we investigate temporal clusters of extremes defined as subsequent exceedances of high thresholds in a stationary time series. Two meaningful features of these clusters are the probability distribution of the cluster size and the ordinal patterns within a cluster. Since these patterns take only the ordinal structure of consecutive data points into account the method is robust under monotone transformations and measurement errors. We verify the existence of the corresponding limit distributions in the framework of regularly varying time series, develop non-parametric estimators and show their asymptotic normality under appropriate mixing conditions. The performance of the estimators is demonstrated in a simulated example and a real data application to discharge data of the river Rhine.