Bounds for a sum of random variables under a mixture of normals

In two papers: Dhaene et al. (2002). Insurance: Mathematics and Economics 31, pp.3-33 and pp. 133-161, the approximation for sums of random variables (rv’s) was derived for the case where the distribution of the components is lognormal and known, but the stochastic dependence structure is unknown or too cumbersome to work with. In finance and actuarial science a lot of attention is paid to a regime switching model. In this paper we give the approximation for sums under a mixture of normals and consider approximate evaluation of provision under switching regime.