Non‐asymptotic moment bounds for random variables rounded to non‐uniformly spaced sets

We study the effects of rounding on moments random variables. Specifically, given a variable X and its rounded counterpart rd(X), we study|E[Xk]-E[rd(X)k]|for non-negative integer k. consider case that function rd: R → F corresponds either to (i) nearest point in some discrete set or (ii) randomly larger smaller this same with probabilities proportional distances these points. In both cases, show, under reasonable assumptions density X, how compute constant C such that|E[Xk] - E[rd(X)k]|< ε2, provided|rd(x) – x|≤ ε E(x), where E: R≥0 is fixed positive piecewise linear function. Refined bounds for absolute E [|Xk rd(X)k|] are also given.