The tail behavior of randomly weighted sums of dependent random variables

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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Published 9 September 2014