On moment convergence for some order statistics

By exploring the uniform integrability of a sequence of some order statistics (OSs), we obtain the moment convergence conclusion of the sequence under some weak conditions even when the corresponding population of interest has no moment of any positive order. As an application, we embody the range of applications of a theorem presented in a reference dealing with the approximation of the difference between the moment of a sequence of normalized OSs and the corresponding moment of a standard normal distribution. By the aid of the embodied theorem, we explore the infinitesimal type of the moments of errors when we estimate some population quantiles by relative OSs. Finally, by the obtained conclusion, we can easily get a combination formula which seems hard to be proved in other methods.