A uniform law of large numbers and empirical central limit theorem for limits of finite populations
Let \(U_ k\) be a finite population of size \(N_ k\) with elements \(a_{kj}\), \(1\leq j\leq N_ k\), all contained in a space U, and let \((Y_{k1},...,Y_{k,N_ k})\) be one of the \(N_ k!\) permutations of \((a_{k1},...,a_{k,N_ k})\) chosen with equal probabilities. The vector \((Y_{k1},...,Y_{k,n_ k})\), \(n_ k\leq N_ k\), is an (ordered) simple random sample from the population \(U_ k\). Define \[ \nu_ k=N_ k^{-1}\sum^{N_ k}_{j=1}\delta_{a_{kj}},\quad \bar Y_ k=n_ k^{-1}\sum^{n_ k}_{j=1}\delta_{Y_{kj}}, \] and \[ \eta_ k(f)=(n_ k^{-1}-N_ k^{-1})^{-1/2}(Y_ k(f)- \nu_ k(f)) \] for \(f\in F\), where F is a family of functions on U. The author gives sufficient conditions with respect to the complexity of F, to ensure that the followings hold. (1) P[\(\sup_{f\in F}| \bar Y_ k(f)-\nu_ k(f)| >\epsilon_ k]\to 0\) as \(k\to \infty\), where \(\epsilon_ k\) is a sequence of positive numbers tending to 0. (2) \(\nu_ k\) converges to a mean zero F-indexed Gaussian process G with covariance \(E(G(f)G(g))=(\nu (fg)-\nu (f)\nu (g)),\) f,g\(\in F\), where \(\nu\) is the weak limit of \(\{\nu_ k\}.\) These generalize results of \textit{B. Rosén} [Ark. Mat. 5, 383-424 (1965; Zbl 0127.105)].
scientific article; zbMATH DE number 4174017
scientific article