Before revision:  In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |E[X^k] - E[rd(X)^{k}]| < C\epsilon^2, provided  \operatorname{rd}(x) - x| \leq \epsilon  E(x) . Asymptotic and non-asymptotic }\to _{\geq 0} is some fixed positive piecewise linear function.
After revision:  In both cases, we show, under reasonable assumptions on the density function of X, how to compute a constant C such that |E[X^k] - E[rd(X)^{k}]| < C\epsilon^2, provided \operatorname{rd}(x) - x| \leq \epsilon \: E(x) . Asymptotic and non-asymptotic }\to _{\geq 0} is some fixed positive piecewise linear function.