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Abstract Let X1,X2,...be a sequence of dependent and heavy-tailed random variables with distributions F1,F2,…. on (-∞,∞),and let т be a nonnegative integer-valued random variable independent of the sequence {Xk,k≥ 1}.In this framework,the asymptotic behavior of the tail probabilities of the quantities Sn =n∑k=1 Xk and S(n) =max1≤k≤n Sk for n ＞ 1,and their randomized versions Sт and S(т) are studied.Some applications to the risk theory are presented.