On the weak invariance principle for random fields with commuting filtrations under L1-projective criteria
Christophe Cuny, Jérôme Dedecker, Florence Merlevède
TL;DR
This paper develops ${\mathbb L}^1$-projective criteria for the weak invariance principle and central limit theorem for random fields indexed by ${\mathbb Z}^d$ with commuting filtrations, under the assumption that at least one generating transformation is ergodic. The authors establish an ${\mathbb L}^2$-orthomartingale approximation when a summability condition involving tail/inverse functions holds, and deduce CLTs with variance $\sigma^2(f)=\sum_{\mathbf{k}}\mathrm{Cov}(X_{\mathbf{0}},X_{\mathbf{k}})$. They extend these results to completely commuting transformations (with reverse orthomartingale decompositions) and provide functional CLTs (weak invariance principle) via a new maximal inequality, yielding convergence to a Brownian sheet under explicit modulus-of-continuity or tail conditions. The theory is illustrated through linear random fields and functions of expanding endomorphisms of the torus, including explicit modulus-based criteria for the toral setting. Overall, the work broadens CLT/FCLT applicability to dependent random fields with L1-type projective control, enabling applications to dynamical systems and spatial models with commuting dynamics.
Abstract
We consider a field $f \circ T_1^{i_1} \circ \cdots \circ T_d^{i_d}$ where $T_1, \dots , T_d$ arecommuting transformations, one of them at least being ergodic. Considering the case of commuting filtrations, we are interested by giving sufficient ${\mathbb L}^1$-projective conditions ensuring that the normalized partial sums indexed by quadrants converge in distribution to a normal random variable. We also give sufficient conditions ensuring the weak invariance principle for the partial sums process. For the central limit theorem (CLT), the proof combines a truncated orthomartingale approximation with the CLT for orthomartingales due to Voln{ý}. For the functional form, a new maximal inequality is needed and is obtained via truncation techniques, blocking arguments and orthomartingale approximations. The case of completely commuting transformations in the sense of Gordin can be handled in a similar way. Application to bounded Lipschitz functions of linear fields whose innovations have moments of a logarithmic order will be provided, as well as an application to completely commuting endomorphisms of the m-torus. In the latter case, the conditions can be expressed in terms of the ${\mathbb L}^1$-modulus of continuity of $f$.