Before revision:  With high probability, the maximum discrepancy in integration error is O_d(n^{-1{2}}\log n) for compactly supported P and O_d(n^{-1{2}} (\log n)^{d+1\log\log n}) for sub-exponential \mathbb{P}  .
After revision:  With high probability, the maximum discrepancy in integration error is O_d(n^{-1{2}}\log n) for compactly supported P and O_d(n^{-1{2}} (\log n)^{d+1\log\log n}) for sub-exponential \mathbb{P} on \mathbb{R .