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In an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic. We will determine approximations for sums of random variables, when the distributions of the terms are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. In this paper, the theoretical aspects are considered. Applications of this theory are considered in a subsequent paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

The optical absorption of polarons at rest at zero temperature is calculated starting from the Feynman-Hellwarth-Iddings-Platzman (FHIP) theory of the impedance.The results are compared with the results of theories whose physical interpretation is clearer [weak-coupling theory of Gurevich, Lang, and Firsov (GLF) and product-ansatz strong-coupling theory of Kartheuser, Evrard, and Devreese (KED)] in order. to obtain a better understanding of the FHIP approximation.We are particularly interested in the possible role of lattice relaxation [leading to relaxed excited states {RES)j in the optical absorption process. If the FHIP perturbation method were used to expand the conductivity {this would be the normal procedure), essentially Franck-Condon transitions would be found in the spectrum, and lattice relaxation would be absent. In this case the results do not fit with the product ansatz and provide merely the asymptotic limit & 0, where 0' is the electron-phonon coupling constant. If, however, the impedance function rather than the conductivity is expanded (as preferred by FHIP for intuitive reasons, without further justification) more reliable results for the optical absorption appear. For G' &5, intense absorption peaks now occur, which presumably correspond to transitions to RES, and the results are in qualitative agreement with the predictions of the product-ansatz treatment in this coupling region. Also in the limit e -0 the correct behavior is found.For 3 & e & 5, the interpretation of the results is somewhat delicate but the possibility that RES contribute to the oscillator strength as soon as 0: &3 should be considered. The results so obtained for the optical absorption seem reliable at all &. This provides an indirect justification for the expansion of Z{Q) rather than 1/Z{Q) in FHIP theory and a confirmation of the qualitative strong-coupling predictions of KED. The present study indicates that optical absorption peaks due to free polarons should be observable experimentally in crystals for which a &1.

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In this contribution, the upper bounds for sums of dependent random variables Xl + X 2 + ... + Xn derived by using comonotonicity are sharpened for the case when there exists a random variable Z such that the distribution functions of the Xi, given Z = z, are known. By a similar technique, lower bounds are derived. A numerical application for the case of lognormal random variables is given.

In an insurance context, one is often interested in the distribution function of a sum of random variables. Such a sum appears when considering the aggregate claims of an insurance portfolio over a certain reference period. It also appears when considering discounted payments related to a single policy or a portfolio at different future points in time. The assumption of mutual independence between the components of the sum is very convenient from a computational point of view, but sometimes not realistic. In Dhaene, Denuit, Goovaerts, Kaas, Vyncke (2001), we determined approximations for sums of random variables, when the distributions of the components are known, but the stochastic dependence structure between them is unknown or too cumbersome to work with. Practical applications of this theory will be considered in this paper. Both papers are to a large extent an overview of recent research results obtained by the authors, but also new theoretical and practical results are presented.

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