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Asymptotics for Tail Probabilities of the Sum and Its Maximum of Extended Negatively Dependent and Heavy-Tailed Random Variables

Let $X_1,X_2,\ldots,X_n$ be a sequence of extended negatively dependent random variables with distributions $F_1,F_2,\ldots,F_n$,respectively. Denote by $S_n=X_1+X_2+\cdots+X_n$. This paper establishes the asymptotic relationship for the quantities $\pr(S_n>x)$, $\pr(\max\{X_1,X_2, \ldots,X_n\}>x)$, $\pr(\max\{S_1,S_2$, $\ldots,S_n\}>x)$ and $\tsm_{k=1}^n\pr(X_k>x)$ in the three heavy-tailed cases. Based on this, this paper also investigates the asymptotics for the tail probability of the maximum of randomly weighted sums, and checks its accuracy via Monte Carlo simulations. Finally, as an application to the discrete-time risk model with insurance and financial risks, the asymptotic estimate for the finite-time ruin probability is derived.