Risk Bounds for Levy Processes in the PAC-Learning Framework

Lévy processes play an important role in the stochastic process theory. However, since samples are non-i.i.d., statistical learning results based on the i.i.d. scenarios cannot be utilized to study the risk bounds for Lévy processes. In this paper, we present risk bounds for non-i.i.d. samples drawn from Lévy processes in the PAC-learning framework. In particular, by using a concentration inequality for infinitely divisible distributions, we first prove that the function of risk error is Lipschitz continuous with a high probability, and then by using a specific concentration inequality for Lévy processes, we obtain the risk bounds for non-i.i.d. samples drawn from Lévy processes without Gaussian components. Based on the resulted risk bounds, we analyze the factors that affect the convergence of the risk bounds and then prove the convergence.