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

When assessing risks on a finite-time horizon, the problem can often be reduced to the study of a random sequence $C(N)=(C_1,\dots ,C_N)$ 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 $\mathrm{\mid }\mathrm{\mid }C(N)\mathrm{\mid }\mathrm{\mid }$, for some norm $\Vert \cdot \Vert$. We propose a generalization of Breiman Lemma that gives way to an asymptotic equivalent to $\mathrm{\parallel }C(N)\mathrm{\parallel }$ 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.