Persistent-Transient Policy Evaluation for Markov Chains via Minimal Peripheral Quotients

We study fixed-policy evaluation for finite Markov chains that may be reducible and periodic. Classical evaluation methods with gain and bias decomposition are not always diagnostic: the gain records only invariant Ces`aro averages, while persistent phase-dependent behavior is absorbed into the bias together with genuinely transient effects. We identify the real peripheral invariant subspace K(P) of the transition matrix P as the source of this ambiguity. Quotienting by K(P) is the minimal exact quotient that removes all non-decaying modes and makes the remaining dynamics strictly stable. After choosing a gauge projection Pi with kernel K(P), the reward admits a unique decomposition r = g_Pi^star + (I-P)v_Pi^star, where g_Pi^star is a persistent regime profile and v_Pi^star is a gauge-fixed transient component. An exact comparison with classical normalized gain and bias shows that the new pair reallocates the same information so that all persistent modes are represented in g_Pi^star and v_Pi^star is transient. This decomposition reconstructs finite-horizon returns, recovers statewise average reward, admits a transient-cost interpretation, and yields a stable estimator under a generative model.