[Paper Review] A New Condition for the Invariance Principle for Stationary Random Fields

Yizao Wang, Michael Woodroofe|arXiv (Cornell University)|Jan 27, 2011

Financial Risk and Volatility Modeling29 references3 citations

TL;DR

This paper establishes a central limit theorem and invariance principle for stationary random fields in multidimensional settings using an $m$-dependent approximation method, introducing a projective-type condition analogous to the Maxwell–Woodroofe condition. The key contribution is extending such invariance principles to random fields, overcoming limitations of martingale approximation in higher dimensions.

ABSTRACT

We establish a central limit theorem and an invariance principle for stationary random fields, with projective-type conditions. Our result is obtained via an m-dependent approximation method. As applications, we establish invariance principles for orthomartingales and functionals of linear random fields.

Motivation & Objective

• To extend the invariance principle and central limit theorem from one-dimensional stationary processes to multidimensional stationary random fields.
• To address the challenge that traditional martingale approximation methods fail to generalize effectively to random fields in dimensions $d \geq 2$.
• To develop a new condition—similar in form to the Maxwell–Woodroofe condition—for the invariance principle in the context of random fields.
• To establish sufficient conditions under which the normalized partial sums of a stationary random field converge weakly to a Brownian motion in $C[0,1]^d$.

Proposed method

• Utilizes an $m$-dependent approximation method as an alternative to martingale approximation, which is less effective in higher-dimensional random fields.
• Applies a projective-type condition involving conditional expectations: $\sum_{k=1}^\infty \frac{\|\mathbb{E}(f \circ T^k \mid \mathcal{F}_0)\|_2}{k^{1/2}} < \infty$, adapted to the multidimensional setting.
• Employs a coupling technique involving independent copies of negative parts of increments to control the error in the $m$-dependent approximation.
• Uses moment bounds and the Cauchy–Schwarz inequality to control the $L^p$-norms of the approximation error terms.
• Applies a moment inequality from Wu (2002) to bound the moments of functionals of the random field over rectangular blocks.
• Establishes conditional independence properties via the theory of regular conditional probabilities to justify key steps in the error decomposition.

Experimental results

Research questions

• RQ1 Can the invariance principle for stationary processes, known under the Maxwell–Woodroofe condition, be extended to stationary random fields in $\mathbb{Z}^d$ for $d \geq 2$?
• RQ2 Is the $m$-dependent approximation method viable for proving invariance principles in multidimensional random fields where martingale approximation fails?
• RQ3 What projective-type condition on conditional expectations ensures the functional central limit theorem for random fields in higher dimensions?
• RQ4 How can the error in approximating a general stationary random field by an $m$-dependent field be controlled in $L^p$-norm?

Key findings

• The paper establishes a functional central limit theorem for stationary random fields under a projective-type condition: $\sum_{k=1}^\infty \frac{\|\mathbb{E}(f \circ T^k \mid \mathcal{F}_0)\|_2}{k^{1/2}} < \infty$, analogous to the Maxwell–Woodroofe condition.
• The normalized partial sum process $\frac{S_{\lfloor n \cdot \rfloor}}{\sqrt{n}}$ converges weakly in $C[0,1]^d$ to a Brownian motion scaled by $\sigma$, where $\sigma^2 = \lim_{n \to \infty} \mathbb{E}(S_n^2)/n$.
• The $m$-dependent approximation method successfully overcomes the limitations of martingale approximation in $d \geq 2$, enabling the derivation of the invariance principle.
• The error in the $m$-dependent approximation is shown to be $O(A_{k+1-h,l+1-h}^{\alpha/2})$ in $L^p$-norm, where $A_{k,l}$ denotes the area of a rectangular block.
• The result applies to orthomartingales and functionals of linear random fields, extending known one-dimensional results to the multidimensional case.
• The condition is shown to be sharp in the sense that it matches the best-known conditions in the one-dimensional case, suggesting its optimality in the multidimensional setting.