Regular variation of a random length sequence of random variables and application to risk assessment

TL;DR

This paper develops a regular variation framework for random sequences of random length, enabling the analysis of risk measures like ruin probability and tail index in applications such as insurance and dietary risk assessment.

Contribution

It introduces a novel regular variation approach for random length sequences, generalizes Breiman's Lemma, and applies these results to risk assessment models.

Findings

Provides asymptotic equivalents for norms of random sequences of random length.

Develops risk indicators such as ruin probability and tail index.

Demonstrates applicability in dietary risk and insurance models.

Abstract

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence $C(N)=(C_1,\ldots,C_N)$ of random length $N$, where $C(N)$ comes from the product of a matrix $A(N)$ of random size $N \times N$ and a random sequence $X(N)$ of random length $N$. Our aim is to build a regular variation framework for such random sequences of random length, to study their spectral properties and, subsequently, to develop risk measures. In several applications, many risk indicators can be expressed from the asymptotic behavior of $\vert \vert C(N)\vert\vert$, for some norm $\Vert \cdot \Vert$. We propose a generalization of Breiman Lemma that gives way to an asymptotic equivalent to $\Vert C(N) \Vert$ and provides risk indicators such as the ruin probability and the tail index for Shot Noise Processes on a finite-time horizon. Lastly, we apply our final result to a model used in dietary risk assessment and in non-life insurance mathematics to illustrate the applicability of our method.