An asymptotic of the distribution of the extrema of independent random variables (Q1907488)

Let \(\{X_n,\;n \geq 1\}\) be a sequence of independent random variables being independent of the sequence of positive integer-valued random variables \(\{N_n,\;n \geq 1\}\). Let \(M(n) = \max(X_1, \dots, X_n)\) and \(V(n) = \min(X_1,\dots,X_n)\). Assume that appropriately normalized \(M(n)\) and \(V(n)\), and the sequence \(\{N_n/n\}\) converges weakly to nondegenerate limited distribution. Under some additional assumptions, this paper exhibits the form of the limiting joint distribution function of appropriately normalized \(M(N_n)\) and \(V(N_n)\), and provides estimates of the rate of convergence. Results here generalize Theorems 2.10.1 and 6.2.1 of \textit{J. Galambos} [``The asymptotic theory of extreme order statistics'' (1978; Zbl 0381.62039)] which provide comparable results for the maxima of a sequence of i.i.d. random variables.