So far the study of exponential bounds of an empirical process has been restricted to abounded index class of functions. The case of an unbounded index class of functions is now studiedon the basis of a new symmetrization idea and a new method of truncating the original probabilityspace; the exponential bounds of the tail probabilities for the supremum of the empirical process overan unbounded class of functions are obtained. The exponential bounds can be used to establish lawsof the logarithm for the empirical processes over unbounded classes of functions.
