Sums of Independent Triangular Arrays and Extreme Order Statistics

Abstract

Let $X_{n,i}$ denote an infinitesimal array of independent random variables with convergent partial sums $Z_n = \sum^n_{i=1} X_{n,i} -a_n \rightarrow_\mathscr{D}\xi$. 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 $(L_n, M_n, W_n) \rightarrow_\mathscr{D} (\xi_1, \xi_2, \xi_3)$ jointly converges for various sequences $k_n, r_n \rightarrow \infty$, where the components of the limit law are independent such that $\xi_1 + \xi_2 + \xi_3 =_\mathscr{D} \xi$. 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.