[Paper Review] A quenched invariance principle for stationary processes

Christophe Cuny, Dalibor Volný|arXiv (Cornell University)|Feb 22, 2012

Stochastic processes and statistical mechanicsMathematics18 references22 citations

TL;DR

This paper establishes a quenched invariance principle for stationary processes under the Hannan condition, proving that conditionally centered partial sums converge weakly to Brownian motion almost surely under the quenched measure. The key contribution is a coupling of the centered sum process with a martingale having stationary increments, ensuring quenched convergence in distribution for the functional central limit theorem under the Hannan condition.

ABSTRACT

In this note, we prove a conditionally centered version of the quenched weak invariance principle under the Hannan condition, for stationary processes. In the course, we obtain a (new) construction of the fact that any stationary process may be seen as a functional of a Markov chain.

Motivation & Objective

• To establish a quenched weak invariance principle for stationary processes under the Hannan condition, extending the quenched central limit theorem to functional convergence.
• To show that the centered sum process $ \bar{S}_n = S_n - \mathbb{E}_0(S_n) $ satisfies the quenched weak invariance principle under the Hannan condition.
• To provide a new construction showing that any stationary process can be represented as a functional of a Markov chain via the conditional expectation operator $ Q $.
• To demonstrate that the quenched convergence holds for the normalized path $ \bar{S}_n(t)/\sqrt{n} $ to a standard Brownian motion under $ \mu $-a.s. quenched measures.

Proposed method

• Use of the conditional expectation operator $ Qh = \mathbb{E}_0(h \circ \theta) $ to define a Markov transition kernel and construct a stationary Markov chain representation of the underlying process.
• Decomposition of the centered sum $ \bar{S}_n(f) $ into a series of martingale increments via $ f_i = P_0(U^i f) $, leveraging the spectral properties of the shift $ \theta $.
• Application of the Doob maximal inequality conditionally to control $ \mathbb{E}_0(\max_{1 \leq n \leq N} \bar{S}_n^2) $, leading to $ L^2 $-type bounds in the quenched setting.
• Use of the Dunford-Schwartz ergodic theorem and weak $ L^2 $-space estimates to control the maximal function of $ (f_i^2)^* $, ensuring integrability under $ \mu $-a.s. convergence.
• Construction of a martingale $ (M_n) $ such that $ \mathbb{E}_0(\max_{1 \leq n \leq N} (\bar{S}_n - M_n)^2) = o(N) $ $ \mu $-a.s., proving asymptotic equivalence in the quenched sense.
• Proof of quenched weak convergence via the convergence of finite-dimensional distributions and tightness, relying on the coupling with a martingale and the continuous mapping theorem.

Experimental results

Research questions

• RQ1 Does the quenched weak invariance principle hold for stationary processes satisfying the Hannan condition?
• RQ2 Can the centered sum process $ \bar{S}_n $ be approximated by a martingale in the quenched $ L^2 $-sense?
• RQ3 Is it possible to represent any stationary process as a functional of a stationary Markov chain via the conditional expectation operator $ Q $?
• RQ4 Under what conditions does the quenched functional central limit theorem hold for $ \bar{S}_n(t)/\sqrt{n} $?
• RQ5 How does the operator $ Q $ facilitate the construction of a Markovian representation of stationary processes?

Key findings

• The quenched weak invariance principle holds: for $ \mu $-a.s. $ x $, the quenched law of $ \bar{S}_n(t)/\sqrt{n} $ converges weakly to that of a standard Brownian motion $ \sigma W_t $.
• The centered sum process $ \bar{S}_n $ satisfies $ \mathbb{E}_0(\max_{1 \leq n \leq N} (\bar{S}_n - M_n)^2) = o(N) $ $ \mu $-a.s., proving asymptotic equivalence with a martingale $ (M_n) $ with stationary ergodic increments.
• The limit variance $ \sigma^2 = \lim_{n \to \infty} \mathbb{E}(S_n^2)/n $ exists and is finite $ \mu $-a.s. under the Hannan condition.
• The condition $ \sum_{n \geq 1} \frac{\|\mathbb{E}_0(X_n)\|_2}{\sqrt{n}} < \infty $ implies the Hannan condition and ensures quenched WIP for $ S_n(t) $, not just $ \bar{S}_n(t) $.
• A new Markov chain representation is constructed via the operator $ Q $, showing that every stationary process arises as a functional of a stationary, time-homogeneous Markov chain with transition kernel $ Q(x, dy) = \mu(x, dy) $.
• The maximal function $ ((f_i^2)^*)^{1/2} $ belongs to the weak $ L^2 $-space $ L^{2,w} $, which ensures the almost sure finiteness of the quenched maximal $ L^2 $-norm of $ \bar{S}_n $.