# The long-term behavior of number of near-maximum insurance claims

## TL;DR

This paper studies the long-term asymptotic behavior of the count of near-maximum insurance claims in a stationary sequence, providing general limit results and corollaries for sums of such claims.

## Contribution

It introduces a general framework for analyzing the asymptotics of near-maximum claims in stationary sequences, extending previous work to broader settings.

## Key findings

- Asymptotic limits for normalized counts of near-maximum claims.
- Results applicable when the index of near-maximum claims grows slower than the total number of claims.
- Corollaries for sums of near-maximum claims in the asymptotic regime.

## Abstract

A near-maximum insurance claim is one falling within a distance $a$ of the current maximal claim. In this paper, we investigate asymptotic behavior of normalized numbers of near-maximum insurance claims under the assumption that the sequence of successive claim sizes forms a strictly stationary process. We present the results in a general form expressing limiting properties of normalized numbers of insurance claims that are in a left neighborhood of the $m_n$th largest claim, where $m_n/n$ tends to zero and $n$ is the number of registered claims. We also give corollaries for sums of near-maximum insurance claims.