## The Annals of Probability

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176988482

**Digital Object Identifier**

doi:10.1214/aop/1176988482

**Mathematical Reviews number (MathSciNet)**

MR1331203

**Zentralblatt MATH identifier**

0836.60012

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60E07: Infinitely divisible distributions; stable distributions

Secondary: 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

#### Citation

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