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

Goovaerts, Marc J.; Kaas, Rob; Laeven, Roger J. A.; Tang, Qihe; Vernic, Raluca

Scand. Actuar. J. 2005, No. 6, 446-461 (2005).

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

Reviewer: A. D. Borisenko (Kyïv)

MSC:

91B30

Risk theory, insurance (MSC2010)

Keywords:

asymptotic; (log)elliptical distribution; (log)normal variance-mean mixed distribution; Pareto-like distribution; tail probability

References:

[1]	DOI: 10.1111/1467-9469.t01-1-00045 · Zbl 0934.62109
[2]	Beirlant J., Practical analysis of extreme values (1996) · Zbl 0888.62003
[3]	Bingham N.H., Regular variation (1987)
[4]	DOI: 10.1088/1469-7688/3/6/302
[5]	DOI: 10.1137/1110037
[6]	DOI: 10.1016/0304-4149(94)90113-9 · Zbl 0799.60015
[7]	DOI: 10.1214/aoap/1035463328 · Zbl 0879.60053
[9]	Fang K.T., Symmetric multivariate and related distributions (1990) · Zbl 0699.62048
[10]	DOI: 10.1007/978-94-011-1646-6
[11]	DOI: 10.1016/S0304-4149(98)00103-3 · Zbl 0962.60075
[12]	DOI: 10.1016/S0304-4149(99)00030-7 · Zbl 0997.60041
[13]	DOI: 10.2307/2328079
[14]	DOI: 10.1080/15326349108807204 · Zbl 0747.60062
[15]	Tang Q., Stochastic Processes and their Applications 108 pp 299– (2003)
[16]	DOI: 10.1239/aap/1103662967 · Zbl 1095.91040
[17]	DOI: 10.1093/rfs/hhg015