On the asymptotic distributions of weighted uniform mulitivariate empirical processes (Q753321)

Let \(F_ n(t,s)\) (0\(\leq t,s\leq 1)\) be an empirical distribution function obtained from a sample of i.i.d. random vectors uniformly distributed on [0,1]\(\times [0,1]\). It is well-known that the process \(\alpha_ n(t,s)=n^{1/2}(F_ n(t,s)-ts)\) can be approximated by a Brownian bridge. The author claims that for small t and s a Poisson process approximates \(\alpha_ n(t,s)\) much better than the Brownian bridge. In fact he considers \[ A_ n=\sup_{t\leq \epsilon_ n,s\leq \delta_ n}| \alpha_ n(t,s)| /t^{\nu}s^ uG(t)L(s) \] where \(\epsilon_ n\searrow 0\), \(\delta_ n\searrow 0\), \(\nu <<\mu <1\) and L and G are slowly varying functions. He proves that \[ r_ nA_ n\to_{D}\sup_{0<t\leq 1,0<s<\infty}| N(t,s)-ts| t^{- \nu}s^{-u} \] where N(.,.) is a Poisson process and \(r_ n\) is a given function of the parameters of \(A_ n\).