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Invariance properties of limiting point processes and applications to clusters of extremes

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Anja Jan{ss}en, Johan Segers

Probability · Submitted 22 Jul 2022

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

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arXiv ID2207.11204 (version 3)
Pith ID2UQBWYPV, Pith's permanent record of this paper and its review
CitationAnja Jan{\ss}en, Johan Segers. "Invariance properties of limiting point processes and applications to clusters of extremes." arXiv:2207.11204 (2022). https://pith.science/paper/2UQBWYPV
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The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,674 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.35 · human participants 0.02 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.46 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v3. Results and the main claim are underlined; .

Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. contribution 0.97 0.87 0.06 asserted 0.99
s1 We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. contribution 0.97 0.85 0.13 asserted 0.98
s2 Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. contribution 0.99 0.94 0.13 asserted 1.00
s3 While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory. result 0.57 0.82 0.43 asserted 0.99

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

Motivated by examples from extreme value theory we introduce the general notion of a cluster process as a limiting point process of returns of a certain event in a time series. We explore general invariance properties of cluster processes which are implied by stationarity of the underlying time series under minimal assumptions. Of particular interest are the cluster size distributions, where we introduce the two notions of inspected and typical cluster sizes and derive general properties of and connections between them. While the extremal index commonly used in extreme value theory is often interpreted as the inverse of a "mean cluster size", we point out that this only holds true for the expected value of the typical cluster size, caused by an effect very similar to the inspection paradox in renewal theory.

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