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We show that for functions f ∈ L p ([0, 1] d), where 1 ≤ p ≤ ∞, the family of integrals [0,x] f (t)dt (x = (x 1 ,. .. , x d) ∈ [0, 1] d) can be approximated by a randomized algorithm uniformly over x ∈ [0, 1] d with the same rate n −1+1/ min(p,2) as the optimal rate for a single integral, where n is the number of samples. We present two algorithms, one(More)
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