On the quenched functional CLT for stationary random fields under projective criteria

The paper’s main rectangle result (Theorem 2.4) proves a quenched functional CLT to a Brownian sheet under the commuting-filtration framework, one ergodic direction, and the Hannan-type projective summability in the Orlicz scale Φ(x)=x^2(log(1+|x|))^{d−1} (with random centering), and identifies the variance via the limit of |n|^{-1}E[S̄_n^2] . Its proof proceeds by a martingale–coboundary/orthomartingale approximation, a maximal estimate derived from a projective decomposition and Cairoli’s inequality, and then invoking the quenched invariance principle for the orthomartingale part together with a multi-parameter functional convergence criterion (Neuhaus) . The candidate solution reproduces the same architecture: it constructs an orthomartingale approximation from projections P0(Xu), controls boundary/tail errors using Cairoli-type bounds and Orlicz summability, and then applies a quenched FCLT for the orthomartingale part plus standard multiparameter tightness, concluding with the same variance identification. Minor differences (e.g., D0 := ∑j∈Zd P0(Xj) in the paper versus d0 := ∑u≥0 P0(Xu) in the model, and Neuhaus vs. Bickel–Wichura for tightness) are cosmetic; the arguments are materially the same and rest on the same estimates and approximations .