In this presentation we provide rates at which strongly consistent estimators
in the sample average approximation approach (SAA) converge to their deterministic
counterparts. To be able to quantify these rates at which convergence
occurs in the almost sure sense, we consider the law of the iterated logarithm in
a Banach space setting. We first establish convergence
rates for the approximating objective functions under relatively mild assumptions.
These rates can then be transferred to the estimators for optimal values and solutions of the approximated problem.
Based on these results, we further show that under the same assumptions
the SAA estimators converge in mean to their deterministic equivalents, at
a rate which essentially coincides with the one in the almost sure sense. Eventually,
we address the notion of convergence in probability and provide some weak convergence rates, albeit under very mild conditions.