A central limit theorem under metric entropy with \(L_ 2\) bracketing
Let (S,\(\rho)\) be a separable pseudo-metric space and let \({\mathcal S}\) denote its Borel \(\sigma\)-field. The triple (V,\({\mathcal V},\mu)\) stands for a probability which is \({\mathcal S}\times {\mathcal V}\)-measurable and suppose that E f(s,V)\(=0\), and E \(f^ 2(s,V)<\infty\) for all \(s\in S\), and \[ (E(f(s,V)-f(t,V))^ 2)^{1/2}\leq \rho (s,t);\quad s,t\in S. \] Assume that \(\{X_ k,k\geq 1\}\) is a sequence of i.i.d. random variables and put \[ S_ n(s)=n^{-1/2}\sum^{n}_{k=1}f(s,X_ k),\quad s\in S. \] Let \(\{\) Z(s),s\(\in S\}\) be a mean zero Gaussian process with \[ Cov(Z(s),Z(t))=Cov(f(s,V_ 1),f(t,V_ 1)). \] The main result states that under the metric entropy integrability condition \(S_ n\) weakly converges to Z as \(n\to \infty\). (For definitions see the paper). Invariance principles and a law of the iterated logarithm are also considered.
scientific article; zbMATH DE number 4088623
scientific article