Replace the double limit in the ACVaR definition with a single limit

Determine whether the Asymptotic CVaR (ACVaR) definition for a finite-state Markov chain with reward function g, given by lim_{m→∞} lim_{n→∞} E[ (1/m) ∑_{k=0}^{m-1} g(X_k) | (1/n) ∑_{k=0}^{n-1} g(X_k) ≥ F^{-1}(c) ], can be replaced by the single-limit expression lim_{n→∞} E[ (1/n) ∑_{k=0}^{n-1} g(X_k) | (1/n) ∑_{k=0}^{n-1} g(X_k) ≥ F^{-1}(c) ] by setting m = n.

Background

The paper defines ACVaR by conditioning on the empirical average of Markovian rewards exceeding a threshold determined by the stationary inverse CDF F^{-1}(c), and uses two separate limits (first in n for the conditioning event, then in m for the averaged quantity) to align with large-deviations-based conditioning results.

This two-limit structure is essential to the theoretical framework borrowed from Borkar–Juneja–Kherani (2004). The authors explicitly ask whether this can be simplified to a single limit by identifying m with n, which would make the measure closer in spirit to classical CVaR and potentially simplify analysis and computation.