Use empirical inverse CDF F_n^{-1}(c) in the ACVaR conditioning threshold

Determine whether the ACVaR definition can replace the stationary inverse CDF threshold F^{-1}(c) with the empirical inverse CDF F_n^{-1}(c), making the threshold part of the limiting process, i.e., whether lim_{m→∞} lim_{n→∞} E[ (1/m) ∑_{k=0}^{m-1} g(X_k) | (1/n) ∑_{k=0}^{n-1} g(X_k) ≥ F_n^{-1}(c) ] is a valid formulation.

Background

In the ACVaR construction, the threshold F^{-1}(c) is taken from the stationary distribution of the reward, estimated via kernel density methods as a preprocessing step. The authors consider whether this threshold could be updated empirically as F_n^{-1}(c) within the algorithm, so that it enters the limiting procedure directly.

Such a modification might better reflect practical estimation and could potentially change the theoretical underpinnings, requiring compatibility with the large-deviations conditioning framework used in the paper.