The randomized complexity of indefinite integration

We show that for functions f∈Lp([0,1]d)f∈Lp([0,1]d), where 1≤p≤∞1≤p≤∞, the family of integrals ∫[0,x]f(t)dt(x=(x1,…,xd)∈[0,1]d) can be approximated by a randomized algorithm uniformly over x∈[0,1]dx∈[0,1]d with the same rate n−1+1/min(p,2)n−1+1/min(p,2) as the optimal rate for a single integral, where nn is the number of samples. We present two algorithms, one being of optimal order, the other up to logarithmic factors. We also prove lower bounds and discuss the dependence of the constants in the error estimates on the dimension.