A note on strong approximations of multivariate empirical processes
Let \(X_ 1,X_ 2,..\). be an i.i.d. sequence of d-dimensional random vectors with distribution function F. Consider the empirical process \(\alpha_ n(x)=n^{1/2}(F_ n(x)-F(x))\), \(x\in R^ d\) \((F_ n\) is the n-th empirical distribution function). The authors show that a Gaussian process \(\{\Gamma (x,t):x\in R^ d,t>0\}\) can be defined such that \[ E \Gamma (x,n)=0,\quad E \Gamma (x,n)\Gamma (y,m)=(n\wedge m)(F(x\wedge y)-F(x)F(y))\quad (x\wedge y=(\min (x_ j,y_ j))_{1\leq j\leq d})\quad \] \[ and\quad \sup_{x\in R^ d}| n^{1/2}\alpha_ n(x)-\Gamma (x,n)| =^{a.s.}O(n^{-(1/(4d))}(\log n)^{3/2}). \] This improves the error term given by \textit{W. Philipp} and \textit{L. Pinzur} [Z. Wahrscheinlichkeitstheor. Verw. Geb. 54, 1-13 (1980; Zbl 0424.60030)]. The proof is based on a result of \textit{I. S. Borisov} [Sib. Mat. Zh. 23, No.5, 31-41 (1982; Zbl 0524.60057)].
scientific article; zbMATH DE number 4062263
scientific article