Asymptotic distribution of extremes of randomly indexed random variables (Q1979094)

The paper is devoted to investigate conditions of convergence \[ \lim_{n\to\infty}\Pr(\min(X_{\tau_1},\dots,X_{\tau_n})\leq y_*+a_n' y)=1-\exp(-y^\alpha),\tag{1} \] where \(X_i\) are i.i.d. random variables with d.f. \(F\) such that \( \lim_{n\to\infty} nF(y_*+a_ny)=y^\alpha\) and \(\tau_1<\tau_2<\dots\) is a sequence of random indexes. It is shown that if for \(k\) large enough and for all \(j\), \(I\{\tau_k=j\}\) is independent of \(X_1\),\dots, \(X_j\), then (1) holds with \(a_n'=a_n\). A sufficient condition of (1) is formulated in ``asymptotic independence'' terms.