Berry-Esseen bounds for standardized subordinators via moduli of smoothness

We introduce moduli of smoothness techniques to deal with Berry-Esseen bounds, and illustrate them by considering standardized subordinators with finite variance. Instead of the classical Berry-Esseen smoothing inequality, we give an easy inequality involving the second modulus. Under finite third moment assumptions, such an inequality provides the main term of the approximation with small constants, even asymptotically sharp constants in the lattice case. Under infinite third moment assumptions, we show that the optimal rate of convergence can be simply written in terms of the first modulus of smoothness of an appropriate function, depending on the characteristic random variable of the subordinator. The preceding results are extended to standardized Levy processes with finite variance.