Asymptotic properties of conditional value-at-risk estimate for asymptotic negatively associated samples

This article examines the strong consistency of the conditional value-at-risk (CVaR) estimate for asymptotic negatively associated (ANA or ${\rho }^{-}$ $\rho ^{-}$ , for short) random samples under mild conditions. It is demonstrated that the optimal rate can achieve nearly $O(n^{-1/2})$ $O (n^{-1/2})$ under certain appropriate conditions. Furthermore, we present numerical simulations and a real data example to corroborate our theoretical results based on finite samples.