In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X1,X2, . . . , which states that, given distributions \( {F}_{X_1} \),\( {F}_{X_2} \), . . . from some class of heavy-tailed distributions, the distribution of the random maximum X(η) := max{0,X1, . . . , Xη} or random maximum S(η) := max{0, S1, . . . , Sη} belongs to the same class of heavy-tailed distributions. Here, Sn = X1 + · · · + Xn, n ≥ 1, and η is a counting random variable, independent of {X1,X2, . . . }. We provide the conditions for the random max-closure property in the case of classes Open image in new window and Open image in new window.