שקיעה קבועה והתרחבות בשני-זמנים-סטוכסטי

תקציר מקורי באנגליתarXiv:2610.10246v1 Announce Type: new Abstract: Constant-step stochastic approximation generally has a nonzero stationary mean error that persists under time averaging. This paper studies that error for nonlinear two-timescale recursions driven by an exogenous finite-state Markov chain. Under stated smoothness assumptions and conditions on the stationary distribution, we derive a first-order bias expansion whose error bound remains uniform as the slow step size becomes much smaller than the fast step size. Fast-manifold coordinates keep the associated covariance equation regular in this limit. For fast step $\eta$ and slow step $\varepsilon$, the expansion reveals a mixed contribution $\varepsilon^2/\eta$ alongside terms linear in each step size. This dependence matters for bias reduction: