Statistical Properties for Mixing Markov Chains with Applications to Dynamical Systems

The paper proves the large deviations estimate by a moment-generating-function/‘Bernstein trick’ argument that yields a clean tail 8 exp(−c(ε) n) with c(ε)=c ε^{2+1/p} and a threshold n(ε) of order ε^{-1/p}, with explicit constants (Theorem 2.1 and the optimization step) . The candidate solution’s block-skeleton/Azuma approach produces an extra polynomial prefactor m(ε)≈ε^{-1/p} via a union bound over arithmetic progressions; its claim that this factor can be absorbed into the exponential for n(ε)≈ε^{-1/p} is incorrect. Absorbing log m requires n(ε) of order ε^{-2-1/p} log(1/ε), contradicting the paper’s stated n(ε)≈ε^{-1/p}. Hence the paper’s result is correct, while the candidate solution’s final step (constant handling and threshold) is flawed.