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:
\[ (i) ~ 0<\alpha<1 \text{ and } \sum_{k=1}^{\infty}E\theta_{k}^{\alpha+\delta}<\infty \text{ and } \sum_{k=1}^{\infty}E\theta_{k}^{\alpha-\delta}<\infty \text{ for some } \delta>0; \]
\[ (ii) ~ \sum_{k=1}^{\infty}(E\theta_{k}^{\alpha+\delta})^{1/(\alpha+\delta)}<\infty \text{ and } \sum_{k=1}^{\infty}(E\theta_{k}^{\alpha-\delta})^{1/(\alpha+\delta)}<\infty \text{ for some }\delta>0. \]


91B30 Risk theory, insurance (MSC2010)
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