Janssen, Arnold Sums of independent triangular arrays and extreme order statistics. (English) Zbl 0836.60012 Ann. Probab. 22, No. 4, 1766-1793 (1994). Summary: Let \(X_{n,i}\) denote an infinitesimal array of independent random variables with convergent partial sums \(Z_n = \sum^n_{i = 1} X_{n,i} - a_n \to_{\mathcal D} \xi\). Throughout, we find conditions for the convergence of the portion \(k_n\) of lower extremes \(L_n (k_n) = \sum^{k_n}_{i = 1} X_{i : n} - b_n\) given by order statistics \(X_{i : n}\). Similarly, \(W_n (r_n)\) denotes the sum of the \(r_n\) upper extremes and \(M_n = Z_n - L_n - W_n\) stands for the middle part of the sum. It is shown that \((L_n, M_n, W_n) \to_{\mathcal D} (\xi_1, \xi_2, \xi_3)\) jointly converges for various sequences \(k_n\), \(r_n \to \infty\), where the components of the limit law are independent such that \(\xi_1 + \xi_2 + \xi_3 = _{\mathcal D} \xi\). The limit of the middle part \(\xi_2\) is asymptotically normal and \(\xi_1\) \((\xi_3)\) gives the negative (positive) spectral Poisson part of \(\xi\). In the case of a compound Poisson limit distribution we obtain rates of convergence that can be used for applications to insurance mathematics. Cited in 1 ReviewCited in 4 Documents MSC: 60E07 Infinitely divisible distributions; stable distributions 60F05 Central limit and other weak theorems Keywords:infinitely divisible distributions; extreme order statistics; sums of independent random variables; compound Poisson distribution; rate of convergence × Cite Format Result Cite Review PDF Full Text: DOI