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Exact Asymptotics of Bivariate Scale Mixture Distributions

Let (RU_1, R U_2) be a given bivariate scale mixture random vector, with R>0 being independent of the bivariate random vector (U_1,U_2). In this paper we derive exact asymptotic expansions of the tail probability P{RU_1> x, RU_2> ax}, a \in (0,1] as x tends infintiy assuming that R has distribution function in the Gumbel max-domain of attraction and (U_1,U_2) has a specific tail behaviour around some absorbing point. As a special case of our results we retrieve the exact asymptotic behaviour of bivariate polar random vectors. We apply our results to investigate the asymptotic independence and the asymptotic behaviour of conditional excess for bivariate scale mixture distributions.