  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.
