The Annals of Probability

Sums of Independent Triangular Arrays and Extreme Order Statistics

Arnold Janssen

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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


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