Limit theorems for the negative parts of weighted multivariate empirical processes with application (Q1263864)

Let \(X_ i\) be i.i.d. uniform on \(I^ d=[0,1]^ d\), let \[ F_ n(t)=n^{-1}\sum^{n}_{i=1}\prod^{d}_{j=1}I_{[0,t_ j]}(X_{ij}), \] \(t\in I^ d\), be their empirical cdf and, with \(| t| =\prod^{d}_{j=1}t_ j\), \[ U_ n(t)=n^{1/2}(F_ n(t)- | t|), \] \(t\in I^ d\), be the multivariate uniform empirical process. The author studies weak convergence and the a.s. behavior (LIL) for the weighted negative part of \(U_ n(t)\), namely \(U^-_ n(t)/q(| t|)\) where q: [0,1]\(\to (0,\infty)\) is continuous, non- decreasing and positive on [0,1]. These results hold for heavier weights than the corresponding results for \(U_ n\) or \(| U_ n|\). Applications to strong limit theorems for \(| t| /F_ n(t)\) are also given. Exponential inequalities for \[ \sup_{s\leq t}-U_ n(s)\quad and\quad \sup_{\alpha \leq | t| \leq \beta}U^-_ n(t)/q(| t|) \] of the author [Multivariate empirical processes. (1987; Zbl 0619.60031)] are important tools. One of the inequalities is used to give a short proof of Kiefer's exponential inequality for the Kolmogorov-Smirnov statistic of the multivariate empirical process.