On the convergence in the law of sums and maxima of independent random variables with random parameters (Q1907492)

Let \(X_n (t)\), \(t \in T\), \(n \in N\), be stochastic processes on a probability space \((\Omega, {\mathcal A}, P)\) and for each \(n\) let \(X_{n,k}\), \(1 \leq k \leq k_n\), be independent copies of \(X_n\). Denote by \(S_n (t)\) and \(M_n (t)\) the sum and the maximum of \(X_{n,k} (t)\), \(1 \leq k \leq k_n\), resp. Let \(\xi\) and \(\xi_n\), \(n \in N\), be i.i.d. \(T\)-valued random variables defined on another probability space \((\Omega_1, {\mathcal A}_1, P_1)\), and denote by \(S_n^* (\omega_1)\), and \(M_n^* (\omega_1)\), the sum and the maximum of \(X_{n,k} (\xi_k (\omega_1))\), \(1 \leq k \leq k_n\), \(\omega_1 \in \Omega_1\), resp. Assuming the weak convergence of \({\mathcal L} (S_n (t))\) \(({\mathcal L} (M_n (t))\) resp.) for all \(t \in T\), the author gives conditions for the weak convergence of \({\mathcal L} (S_n^* (\omega_1))\) \(({\mathcal L} (M_n^* (\omega_1))\) resp.) for almost all \(\omega_1 \in \Omega_1\). Limit distributions for \(S_n^* (\omega_1)\) and \(M_n^* (\omega_1)\) are described. Some applications to statistics are considered.