Risk Bounds for Infinitely Divisible Distribution
This work provides theoretical risk bounds for statistical learning with infinitely divisible distributions, which is incremental as it extends existing results to a specific distribution class.
The paper tackles the problem of deriving risk bounds for samples from infinitely divisible distributions with zero Gaussian component, using martingale methods to obtain deviation inequalities and covering number-based bounds, and shows that the resulting bound converges faster than previous results for i.i.d. empirical processes.
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).