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

#### Abstract

In this paper, we study the asymptotic behavior of the tail of $X_1+X_2$ in a dependent framework; where $X_1$ and $X_2$ 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 $P(|X_1-X_2|>x) a.P(|X|>X)$ as $x\rightarrow\infty$, where a is a constant and $X_1,X_2$ are dependent random variables.

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