Before revision: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when |  rd (x)-x | \leq \epsilon |x| or | rd (x)-x| \leq \delta which depend linearly on \epsilon and \delta. We then show that if the perturbation corresponds to\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.
After revision: In this paper, we provide general bounds on the mean absolute difference and difference of moments of a random variable X and a perturbation rd (X), when |  rd (x)-x | \leq \epsilon |x| or | rd for non-negative integer k. We consider the case that the rounding function\operatorname{rd} }\to rounding to the nearest point in some fixed discrete set , the bounds on the difference of moments can be improved to quadratic in many cases.