随机长度随机变量序列的正则变化及其在风险评估中的应用
Regular variation of a random length sequence of random variables and application to risk assessment
摘要：
当评估有限时间范围内的风险时，问题通常可以简化为研究随机长度为$N$的随机序列$C(N)=(C_1,\ldots,C_N)$，其中$C(N)$来自随机大小的矩阵$A(N)$ $N \times N$和随机长度为$N$的随机序列$X(N)$的乘积。我们的目标是建立一个定期的变化框架，这样的随机序列的随机长度，研究其频谱特性，随后，开发风险措施。在一些应用中，许多风险指标可以从$\vert \vert C(N)\vert\vert$的渐近行为表示，对于某些范数$\Vert \cdot \Vert$。我们提出了一个推广的Breiman引理，让方式的渐近等价于$\Vert C(N) \Vert$，并提供风险指标，如破产概率和尾部指数有限时间范围内的散粒噪声过程。最后，我们将我们的最终结果应用于饮食风险评估和非人寿保险数学中的模型，以说明我们的方法的适用性。
英文摘要：
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.
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