A note on the approximation of the uniform empirical process
Let \(\{\epsilon_ k\), \(k\geq 1\}\) be a sequence of independent and on the interval [0,1] uniformly distributed random variables and define the empirical distribution function \(F_ n(t)\), \(n\geq 1\), as \[ F(t)=1/n=\{\epsilon_ j,\quad j\leq n,\quad \epsilon_ j\leq t\},\quad 0\leq t\leq 1. \] \(P_ n(t)\), \(0\leq t\leq 1\), stands for a Poisson process with parameter n. Approximations of the empirical process by a Poisson process in a small neighbourhood of zero are considered. The main results are given as follows: Theorem 1. For all \(n\geq 1\), a Poisson process \(P_ n(t)\) with parameter n on the interval [0,1] and a sequence of independent and on the interval uniformly distributed random variables \(\epsilon_ 1,...,\epsilon_ n\) can be constructed simultaneously in such a way that the empirical distribution function defined with the help of these \(\epsilon\) 's satisfies the relation \[ P(\sup_{0<s<n^{-2/3}}| n(F_ n(s)-s)- (P_ n(s)-ns)| >C)<K \exp (-8^{-1}\sqrt{n}\log n) \] with some universal constants \(C>0\) and \(K>0.\) Theorem 2. (a) For all \(n=1,2,...\), and 1/2\(\geq \alpha \geq 0\) a sequence \(\epsilon_ 1,\epsilon_ 2,...,\epsilon_ n\) of independent and on the interval [0,1] uniformly distributed random variables can be constructed together with a sequence \(P_ 1(t),P_ 2(t),...,P_ n(t)\) of independent Poisson processes with parameter 1 on the interval [0,1] in such a way that \[ P(\sup_{k<n}\sup_{0<t<n^{-1/2+\alpha}}| k(F_ k(t)-t)-\sum^{k}_{j=1}(P_ j(t)-t)| >m)\leq C(m)n^{- 2(m+1)\alpha} \] for all \(m=1,2,...\), where the constants C(m) depend only on m. (b) For any \(\delta >0\), an infinite sequence \(\epsilon_ 1,\epsilon_ 2,..\). of independent random variables with uniform distribution on the interval [0,1] and an infinite sequence of independent Poisson processes \(P_ 1(t),P_ 2(t),..\). with parameter 1 on the interval [0,1] can be constructed in such a way that \[ P(\sup_{k}\sup_{0<t<n^{-1/2- \delta}}| n(F_ n(t)-t)-\sum^{n}_{j=1}(P_ j(t)-t)| <\infty)=1. \]
scientific article; zbMATH DE number 4153584