A central limit theorem for certain nonlinear statistics in repeated sampling of a finite population

We prove a central limit theorem for the asymptotic joint distribution of non-linear statistics of the form [summation operator]l = 1N XNltItlIt+1,l, 1 [less-than-or-equals, slant] t [less-than-or-equals, slant] r - 1, and linear statistics of the form [summation operator]l = 1N XNltItl, 1 [less-than-or-equals, slant] t [less-than-or-equals, slant] r, based on independent repeated samples of a finite population of size N with sample indicators Itl,l = 1, ..., N, for the tth sample.