  We study the background risk model under a various forms of dependence and
some distribution classes of heavy tails. First, we study the asymptotic
behavior of tail expectation of a portfolios with unequal heavytailedness, of
non-random sums and randomly stopped sums, under some dependence structure
which contains the independence as a special case. Further we investigate the
asymptotic behavior of a pair of randomly weighted sums, generalizing the
dependence structure among random vector components, our results contains the
finite ruin probability in bi-dimensional discrete time risk model with unequal
heavytailedness. Next, we carry out asymptotic analysis of the tail distortion
risk measures in background risk model under various forms of dependence, with
multivariate regularly varying risk distributions in each portfolio.
