## The tail probability of discount sums of Pareto-like losses in insurance.(English)Zbl 1144.91026

The authors investigate the tail probabilities of the randomly weighted sums $$\sum_{k=1}^{n}\theta_{k}X_{k}$$, $$n=1,2,\dots$$ and their maxima. Here $$\{X_{n}, n=1,2,\dots\}$$ is a sequence of i.i.d. random variables with common distribution function $$F=1-\bar F$$, while $$\{\theta_{n}, n=1,2,\dots\}$$ is a sequence of dependent nonnegative random variables, independent of the sequence $$\{X_{n}, n=1,2,\dots\}$$. The main results of paper are following. Let for $$\alpha>0$$ $$\bar F(x)=x^{-\alpha}L(x)$$, $$x>0$$, where $$L(\cdot)$$ is a slowly varying function. We have $Pr\left(\max_{1\leq m\leq n}\sum_{k=1}^{m}\theta_{k}X_{k}>x\right)\sim Pr\left(\sum_{k=1}^{n}\theta_{k}X_{k}>x\right)\sim\bar F(x)\sum_{k=1}^{n}E\theta_{k}^{\alpha}$ if there exists some $$\delta>0$$ such that $$E\theta_{k}^{\alpha+\delta}<\infty$$ for each $$1\leq k\leq n$$. We have $Pr\left(\max_{1\leq n<\infty}\sum_{k=1}^{n}\theta_{k}X_{k}>x\right)\sim Pr\left(\sum_{k=1}^{\infty}\theta_{k}X_{k}^{+}>x\right)\sim\bar F(x)\sum_{k=1}^{\infty}E\theta_{k}^{\alpha}$ if one of the following assumptions holds:
$(i) ~ 0<\alpha<1 \text{ and } \sum_{k=1}^{\infty}E\theta_{k}^{\alpha+\delta}<\infty \text{ and } \sum_{k=1}^{\infty}E\theta_{k}^{\alpha-\delta}<\infty \text{ for some } \delta>0;$
$(ii) ~ \sum_{k=1}^{\infty}(E\theta_{k}^{\alpha+\delta})^{1/(\alpha+\delta)}<\infty \text{ and } \sum_{k=1}^{\infty}(E\theta_{k}^{\alpha-\delta})^{1/(\alpha+\delta)}<\infty \text{ for some }\delta>0.$

### MSC:

 91B30 Risk theory, insurance (MSC2010)
Full Text:

### References:

  DOI: 10.1111/1467-9469.t01-1-00045 · Zbl 0934.62109  Beirlant J., Practical analysis of extreme values (1996) · Zbl 0888.62003  Bingham N.H., Regular variation (1987)  DOI: 10.1088/1469-7688/3/6/302  DOI: 10.1137/1110037  DOI: 10.1016/0304-4149(94)90113-9 · Zbl 0799.60015  DOI: 10.1214/aoap/1035463328 · Zbl 0879.60053  Fang K.T., Symmetric multivariate and related distributions (1990) · Zbl 0699.62048  DOI: 10.1007/978-94-011-1646-6  DOI: 10.1016/S0304-4149(98)00103-3 · Zbl 0962.60075  DOI: 10.1016/S0304-4149(99)00030-7 · Zbl 0997.60041  DOI: 10.2307/2328079  DOI: 10.1080/15326349108807204 · Zbl 0747.60062  Tang Q., Stochastic Processes and their Applications 108 pp 299– (2003)  DOI: 10.1239/aap/1103662967 · Zbl 1095.91040  DOI: 10.1093/rfs/hhg015
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.