Probability estimates for multiparameter Brownian processes
Let F be a distribution function on \([0,1]^ d\), \(X_ i\), \(i=1,2,..\). independent random vectors with distribution function F, and \(F_ n(t)=n^{-1}\sum^{n}_{i=1}I_{[0,t]}(X_ i)\) be the empirical distribution function. Then \(\sqrt{n}(F_ n(\cdot)-F(\cdot))\) converges in law to the mean zero Gaussian process \(W_ F\) with \[ cov(W_ F(s),\quad W_ F(t))=F([0,s]\cap [0,t])-F([0,s])F([0,t]). \] If G is a function on \([0,1]^ d\), upper and lower bounds are found for \[ P\{\sup_{t\in [0,1]^ d}| W_ F(t)-G(t)| \leq \epsilon \}. \]
scientific article; zbMATH DE number 4052694