We study the asymptotic covariance function of the sample mean and quantile, and derive a new and surprising characterization of the normal distribution: the asymptotic covariance between the sample mean and quantile is constant across all quantiles, if and only if the underlying distribution is normal. This is a powerful result and facilitates statistical inference. Utilizing this result, we develop a new omnibus test for normality based on the quantile-mean covariance process. Compared to existing normality tests, the proposed testing procedure has several important attractive features. Monte Carlo evidence shows that the proposed test possesses good finite sample properties. In addition to the formal test, we suggest a graphical procedure that is easy to implement and visualize in practice. Finally, we illustrate the use of the suggested techniques with an application to stock return datasets.

Recent studies of single-walled carbon nanotubes (CNTs) in aqueous media have showed that water can significantly affect the tube mechanical properties. CNTs under hydrostatic compression can preserve their elastic properties up to large pressure values, while exhibiting exceptional resistance to mechanical loadings. It was experimentally observed that CNTs with encapsulated linear carbon chains (LCCs), when subjected to high hydrostatic pressure values, present irreversible red shifts in some of their vibrational frequencies. In order to address the cause of this phenomenon, we have carried out fully atomistic reactive (ReaxFF) molecular dynamics (MD) simulations for model structures mimicking the experimental conditions. We have considered the cases of finite and infinite (cyclic boundary conditions) CNTs filled with LCCs (LCC@CNTs) of different lengths (from 9 up to 40 atoms). Our results show that increasing the hydrostatic pressure causes the CNT to be deformed in an inhomogeneous way due to the LCC presence. The LCC/CNT interface regions exhibit convex curvatures, which results in more reactive sites, thus favoring the formation of covalent chemical bonds between the chain and the nanotube. This process is irreversible with the newly formed bonds continuing to exist even after releasing the external pressure and causing an irreversibly red shift in the chain vibrational modes from 1850 to 1500 cm−1.