  In this paper, we study the risk bounds for samples independently drawn from
an infinitely divisible (ID) distribution. In particular, based on a martingale
method, we develop two deviation inequalities for a sequence of random
variables of an ID distribution with zero Gaussian component. By applying the
deviation inequalities, we obtain the risk bounds based on the covering number
for the ID distribution. Finally, we analyze the asymptotic convergence of the
risk bound derived from one of the two deviation inequalities and show that the
convergence rate of the bound is faster than the result for the generic i.i.d.
empirical process (Mendelson, 2003).
