Tail behavior of a sum of two dependence and heavy-tailed distributions

We consider the problem of a sum of two dependent and heavy tailed distributions through the C-convolution. The C-convolution provides the distribution of the sum of two random variables whose dependence structure is described by a copula function. Moreover, to investigate the role of heavy tails we use three different marginal distributions characterized by this property: Cauchy, Levy and Pareto. We show that the tail behavior of the C-convolution measured by level-q quantiles for q = 0.01, 0.05 (left tail) and q = 0.95, 0.99 (right tail) is strongly affected by the copula function which links the marginals and by the tail heaviness of marginals themselves.