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

Risk Bounds for Infinitely Divisible Distribution