  The statistical analysis of Randomized Numerical Linear Algebra (RandNLA)
algorithms within the past few years has mostly focused on their performance as
point estimators. However, this is insufficient for conducting statistical
inference, e.g., constructing confidence intervals and hypothesis testing,
since the distribution of the estimator is lacking. In this article, we develop
an asymptotic analysis to derive the distribution of RandNLA sampling
estimators for the least-squares problem. In particular, we derive the
asymptotic distribution of a general sampling estimator with arbitrary sampling
probabilities. The analysis is conducted in two complementary settings, i.e.,
when the objective of interest is to approximate the full sample estimator or
is to infer the underlying ground truth model parameters. For each setting, we
show that the sampling estimator is asymptotically normally distributed under
mild regularity conditions. Moreover, the sampling estimator is asymptotically
unbiased in both settings. Based on our asymptotic analysis, we use two
criteria, the Asymptotic Mean Squared Error (AMSE) and the Expected Asymptotic
Mean Squared Error (EAMSE), to identify optimal sampling probabilities. Several
of these optimal sampling probability distributions are new to the literature,
e.g., the root leverage sampling estimator and the predictor length sampling
estimator. Our theoretical results clarify the role of leverage in the sampling
process, and our empirical results demonstrate improvements over existing
methods.
