On the Glivenko-Cantelli theorem for generalized empirical processes based on strong mixing sequences
Given \(\{X_ i,i\geq 1\}\) as non-stationary strong mixing sequence of r.v.'s with mixing coefficients \(\alpha\) (n), for \(1\leq i\leq n\), and some \(\gamma\in [0,1]\), let \(F_ i(x)=\gamma P(X_ i<x)+(1-\gamma)P(X_ i\geq x)\) and \(I_ i=\gamma I(X_ i<x)+(1-\gamma)I(X_ i\geq x)\). For any sequence \(\{C_ i\}\) satisfying certain conditions, let \[ D_ n=\sup_{x,\gamma}\max_{N\leq n}| \sum^{N}_{i=1}C_ i[I_ i(x)-F_ i(x)]|. \] The main result of this paper is as follows: Theorem. Let \(m=m(n)\), \(r=r(n)\) be two positive integer functions of n such that \(1<m<n\) and \(r=[n/2m]\), [x] being the integer part of x and let m and r tend to infinity as \(n\to \infty\). Let \(\beta =\sum^{n}_{i=1}| C_ i|\) and \(d_ n=\max_{i\leq n}| C_ i|\). Then, for each \(n\geq 1\) and for any \(a_ n<b_ n\) where \(b_ n=2\sqrt{2}d_ n/\sqrt{r},\) (i) \(P(D_ n>a_ n)\leq \beta_ n(ra_ n/n^ 2d^ 2_ n)[\exp \{- 2((a_ n/b_ n)^ 2-1)\}K_{\alpha,n}+Kr\alpha (2m)]\), where \(K_{\alpha,n}=1+K\alpha^{\delta}(m)e^{2r}\) \((1/2\leq \delta <1)\) and (ii) \(D_ n=O(a_ n)\) a.s. for n sufficiently large, provided \[ \sum^{\infty}_{n=1}\beta_ n(a_ nr/n^ 2d^ 2_ n)[\exp (-a^ 2_ nr/4n^ 2d^ 2_ n)\cdot K_{\alpha,n}+Kr\alpha (2m)]<\infty. \]
scientific article; zbMATH DE number 4172051