## 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)
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### References:

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