Asymptotic Behavior of Convolution of Dependent Random Variables with Heavy-Tailed Distributions

In this paper, we study the asymptotic behavior of the tail of X-1 + X-2 in a dependent framework; where Xi and X2 are two positive heavy-tailed random variables with continuous joint and common marginal distribution functions F(x, y) and F(x), respectively; and for some classes of heavy-tailed distributions, we obtain some bounds and convolution properties. Furthermore, we prove P(vertical bar X-1 - X-2 vertical bar > x) similar to a.P(vertical bar X vertical bar > x) as x -> infinity, where a is a constant and X-1, X-2 are dependent random variables.