Suppose that X1,…,Xn are the independent d-dimensional random vectors. In this paper we consider the empirical processes, which are constructed by the random vectors, and indexed by the classes of functions with polynomial discrimination. We use the repeated chain method, randomization method and symmetrization method to derive the exponential bounds which are close to the optimal bounds. As an application, we investigate the Projection Pursuit problems.
ZHU LI-XING,ZHANO JIAN,CHENG PING. THE EXPONENTIAL BOUNDS FOR THE UNIFORM DEVIATION PROBABILITIES OF EMPIRICAL PROCESSES WITH THEIR APPLICATIONS (Ⅰ)[J]. Acta Mathematicae Applicatae Sinica, 1993, 16(4): 544-559.