On Efficient Range-Summability of IID Random Variables in Two or Higher Dimensions

$d$-dimensional efficient range-summability ($d$D-ERS) of a long list of random variables (RVs) is a fundamental algorithmic problem that has applications to two important families of database problems, namely, fast approximate wavelet tracking (FAWT) on data streams and approximately answering range-sum queries over a data cube. In this work, we propose a novel solution framework to $d$D-ERS for $d>1$ on RVs that have Gaussian or Poisson distribution. Our solutions are the first ones that compute any rectangular range-sum of the RVs in polylogarithmic time. Furthermore, we develop a novel $k$-wise independence theory that allows our $d$D-ERS solutions to have both high computational efficiencies and strong provable independence guarantees. Finally, we generalize existing DST-based solutions for 1D-ERS to 2D, and characterize a sufficient and likely necessary condition on the target distribution for this generalization to be feasible.