Asymptotic behavior of random time absolute ruin probability with D∩LD∩L tailed and conditionally independent claim sizes

Consider the probability of random time absolute ruin in the renewal risk model with constant premium rate and constant force of interest. We assume that claim sizes Xi,i=1,2,…Xi,i=1,2,…, are conditionally independent on some sigma algebra and that the common distribution belongs to class D∩LD∩L. We obtain the asymptotic formula for the subclass of subexponential distributions.