Statistics and Its Interface

Volume 7 (2014)

Number 3

Special Issue on Extreme Theory and Application (Part I)

Guest Editors: Yazhen Wang and Zhengjun Zhang

The tail behavior of randomly weighted sums of dependent random variables

Pages: 331 – 338

DOI: http://dx.doi.org/10.4310/SII.2014.v7.n3.a3

Authors

Xuan Leng (Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui, China)

Taizhong Hu (Department of Statistics and Finance, School of Management, University of Science and Technology of China, Hefei, Anhui, China)

Abstract

Consider dependent random variables $X_1, \ldots, X_d$ with a common distribution function $F$ and denote by $\omega_F$ the right endpoint of the support of $F$. Let $\Theta_1, \ldots, \Theta_d$ be non-negative random variables, independent of $ X=(X_1, \ldots, X_d)$ and satisfying certain moment conditions if necessary. Under the assumption that $X$ is in the maximum domain of attraction of a multivariate extreme value distribution, we establish the asymptotic behaviors of randomly weighted sums: there exist limiting constants $q^{\rm F}_{\theta}$, $q^{\rm W}_{\theta}$ and $q^{\rm G}_{ \theta}$ such that for large $t$, $\mathrm{P} (\sum_{i=1}^d \Theta_i X_i > t)\sim\mathrm{E} q^{\rm F}_{\Theta}\cdot \mathrm{P}(X_1 > t)$, $\mathrm{P}(\sum_{i=1}^d \Theta_i(\omega_F-X_i) < 1/t) \sim\mathrm{E} q^{\rm W}_{\Theta}\cdot\mathrm{P} (X_1 >\omega_F-1/t)$, and for $\sum^d_{i=1}\Theta_i=1$ and $t$ approaching to $\omega_F$, $\mathrm{P} (\sum_{i=1}^d \Theta_iX_i >t)\sim\mathrm{E} q^{\rm G}_{\Theta}\cdot\mathrm{P}(X_1 > t)$ according to $F$ belonging to the maximum domains of attraction of the Fréchet, Weibull and Gumbel distributions, respectively. Moreover, some basic properties of the proportionality factor $\mathrm{E} q^{\rm F}_{\Theta}$ are presented.

Keywords

asymptotics, maximum domain of attraction, multivariate extreme value distribution, multivariate regular variation, spectral measure

2010 Mathematics Subject Classification

Primary 60G70. Secondary 62P05.

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