The Annals of Probability

Sums of Independent Triangular Arrays and Extreme Order Statistics

Arnold Janssen

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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.

Article information

Ann. Probab., Volume 22, Number 4 (1994), 1766-1793.

First available in Project Euclid: 19 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60E07: Infinitely divisible distributions; stable distributions
Secondary: 60F05: Central limit and other weak theorems

Infinitely divisible distributions extreme order statistics sums of independent random variables compound Poisson distribution rate of convergence


Janssen, Arnold. Sums of Independent Triangular Arrays and Extreme Order Statistics. Ann. Probab. 22 (1994), no. 4, 1766--1793. doi:10.1214/aop/1176988482.

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