POLYNOMIAL TAIL DECAY FOR STATIONARY MEASURES

The paper proves, for complete metric spaces and finitely supported μ with χ_μ<0, both existence/uniqueness of a stationary measure and a polynomial tail bound, using a pathwise construction plus a large-deviation estimate (Lemma 2.1) and an explicit exponential approximation to stationarity (Lemma 2.2), culminating in Theorem 1.1 and its tail bound ν({y: d(x,y)≥R}) ≪ R^{-α} (see Theorem 1.1 and its proof, including the existence/uniqueness argument and tail derivation via (2.2) and the large deviations step) . The candidate model’s proof correctly identifies a standard drift-plus-Wasserstein contraction strategy: pick α>0 with E[ρ(g)^α]<1; derive a Foster–Lyapunov-type inequality and use a D_α coupling contraction to prove uniqueness and polynomial tails via Markov’s inequality. However, the model’s existence step appeals to Prokhorov’s theorem by claiming that a uniform α-moment bound implies tightness on the (separable) orbit-closure S. On general complete metric spaces, closed balls need not be compact, so bounded α-moments do not by themselves yield uniform tightness; this gap invalidates the existence step as written. The paper avoids this pitfall by constructing the almost-sure limit z(ω)=lim γ_1⋯γ_n x directly and taking its law, which yields stationarity without any tightness or local-compactness assumptions .