Comment on a theorem of M. Maxwell and M. Woodroofe

We present a direct derivation of the theorem of M. Maxwell and M. Woodroofe (Ann. Probab. 28 (2000) 713-724), on martingale approximation of additive functionals of stationary Markov processes, from the non-reversible version of the Kipnis-Varadhan theorem.

This is assumed to be a strongly continuous contraction semigroup, whose infinitesimal generator is denoted by G, which is a well-defined (possibly unbounded) closed linear operator of Hille-Yosida type on H. It is assumed that there exists a dense core C ⊆ H on which G is decomposed as where S is Hermitian and positive semidefinite, while A is skew-Hermitian: Finally, it is assumed that S, respectively, A are essentially self-adjoint, respectively, essentially skew-self-adjoint on the core C. The operator S 1/2 appearing in the forthcoming arguments is defined in terms of the spectral theorem. Let f ∈ H, be such that (f, 1 1) = Ω f dπ = 0, where 1 1 ∈ L 2 (Ω, π) is the constant function 1 1(ω) ≡ 1. We ask about CLT/invariance principle, as N → ∞, for We denote: Recall the non-reversible version of the Kipnis-Varadhan theorem and the theorem of Maxwell and Woodroofe about the CLT problem mentioned above: Theorem KV. With the notation and assumptions as before, if the following two limits hold in H (in norm topology): then exists, and there also exists a zero mean, L 2 -martingale M (t) adapted to the filtration of the Markov process η(t), with stationary and ergodic increments, and variance In particular, if σ > 0, then the finite dimensional marginal distributions of the rescaled process t → σ −1 N −1/2 N t 0 f (η(s)) ds converge to those of a standard 1d Brownian motion. Conditions (1) and (2) of Theorem KV are jointly equivalent to the following Indeed, straightforward computations yield: Theorem MW. With the notation and assumptions as before, if: then the martingale approximation and CLT from Theorem KV hold.
Remarks. • The reversible version (when A = 0) of Theorem KV appears in the celebrated paper [1]. In that case conditions (1) and (2) are equivalent and the proof relies on spectral calculus. The non-reversible formulation of Theorem KV appears -in discrete-time Markov chain, rather than continuous-time Markov process setup and with condition (3) -in [4]. Its proof follows the original proof from [1], with spectral calculus methods replaced by resolvent calculus.
• Theorem MW appears in [3]. Its proof contains elements in common with the arguments of the proof of Theorem KV. However, in the original formulation it's not transparent that Theorem MW is actually a direct consequence of Theorem KV.
• For full historical record of the circle of ideas and results related to Theorem KV (as, e.g., the various sector conditions) and a wide range of applications to tagged particle diffusion in interacting particle systems, random walks and diffusions in random environment, other random walks and diffusions with long memory, etc., see the recent monograph [2].
2 Theorem MW from Theorem KV then conditions (1) and (2) of Theorem KV hold.
Remark. • Proposition 1 also sheds some light on the conditions of Theorem KV: It shows that (1) alone is just marginally short of being sufficient.
The following is essentially Lemma 1 from [3]. We reproduce it only for sake of completeness.
Proof of Lemma 1. This is straightforward computation. Note first that Thus, Next we prove that for any δ ∈ (0, 1) From (11) and (12) the statement of the lemma follows. Fix t ∈ [0, ∞), δ ∈ (0, 1) and denote u k := tδ k . Since the function [0, ∞) ∋ u → u 1/2 e −u is strictly unimodular, there exists a unique k * = k * (t, δ) ∈ Z such that Then the sum on the left hand side of (12) is: Hence (12), and the statement of the lemma follows.