Exponential Bounds for the Uniform Deviation of a Kind of Empirical Processes, II

In this paper, we show that the exponential bounds for the PP Kolmogorov-Smirnov statistic, the uniform deviation of an empirical process indexed by the indicators of some sets based on m-dimensional projections, are c(P) λ(2 + α)(p − 1)m + 2(m − 1) exp(−2λ2), where α (α ≥ 0) and c(P) are constants and P is the population distribution. In particular, α = 0 provided P is an elliptically contoured distribution or some distribution with a bounded support and uniformly bounded marginal density functions with respect to the Lebesgue measure.