Strong approximations for dependent random fields

Introduction

We consider the problem of strong Gaussian approximation for dependent spatial random fields. Concretely, for a centered stationary random field with finite $p$ moments, $(X_i)_{i\in\mathbb{Z}^{d}}$, we prove that there exists (ignoring measure theoretic nuances) an i.i.d Gaussian field $(Y_i)_{i\in\mathbb{Z}^{d}}$ and a constant $\sigma_{\infty}$ such that

where, $S_X(m)=\sum_{i\in A}X_i$, $A=\prod_{j=1}^{d}\{1,\ldots,m_j\}$ with $m_i\le n$ denotes the sum of the random variables $X$ over lower-left rectangular prefixes with top-right corner at $m\in\mathbb{N}^{d}$, and $S_Y(m)$ denotes the corresponding sum over the coupled Gaussian process $Y$. Strong approximation for the time-series case, i.e. $d=1$, has a long history, starting from the Strassen [ [39], Komlós et al. [ [27, 28], Csörgő and Révész [ [16] for i.i.d. observations, with [34, 4, 47, 48, 29, 5, 49, 26, 7] provide corresponding versions in time-series. However, passing from $d=1$ to $d>1$ comes with significant challenges. The main bottleneck can be summarized into the truism that $\mathbb{Z}^{d}$ is only partially ordered. Moreover, control along a growing sequence of cubes does not by itself control every thin rectangle inside them. Finally, the rates for $d=1$ do not provide intuition for the term $(d-1)/2$; for $d=2$ and independent random fields, Rio [ [37] derived the rates $2/p$. However, we show that the rate $\max\{d/p,(d-1)/2\}$ is sharp for dependent random fields; specifically, the term $(d-1)/2$ arises as a cost of dependence.

For independent arrays, Morrow and Philipp [ [33] established set-indexed strong approximations, while Rio [ [35, 37] showed how rates depend on the indexing class. For dependent fields, strong approximations have largely been developed under mixing or association-type conditions [3, 2, 40, 1, 10, 38, 12], while central limit theorems and weak invariance principles are available under mixing, projective, finite-dependence and orthomartingale conditions [Bolthausen, 1982, Dedecker, 1998, 2001, Wang and Woodroofe, 2013, Volný and Wang, 2014, Giraudo, 2018, Cuny et al., 2026]. Our setting departs from this literature in three respects. First, we measure dependence through the functional-dependence coefficients of [47], which can be computed explicitly, as in § 2.3. Mixing and association, in contrast, are properties of the induced $\sigma$-fields or covariances, which are typically hard to verify, and strong mixing can fail even for simple linear processes. Moreover, functional-dependence and spatial Hannan conditions are not interchangeable [43], so results under projective criteria do not automatically cover our setting. Second, we impose no growth restriction on the rectangles, whereas, for instance, Berkes and Morrow [ [3] require growth regions that stay away from the coordinate axes. Third, the closest work in terms of usage of functional dependence measures - El Machkouri et al. [ [21], establishes moment inequalities and central limit theorems for functionals of i.i.d. spatial innovations, but weak limit theorems do not deliver the pathwise rates of a coupling. To the best of our knowledge, obtaining a single, sharp, almost sure Gaussian white-noise coupling for dependent random fields under general enough conditions over unrestricted rectangles, has remained an open problem. We settle this problem with an optimal approximation error.

Our main result, Theorem 2.1 only assumes sufficiently fast decay of spatial dependence along with positive long-run variance. It produces a coupling with an i.i.d Gaussian random field at all observation scales, allowing nonlinear and noncausal fields without requiring strong mixing. The construction uses a permanent dyadic partition to obtain independent surrogate blocks with the correct marginal laws. The spatial versions of Doob or Burkholder-type maximal inequalities from El Machkouri et al. [ [21] and Móricz [ [30] control this replacement error. Within each block, tensor Haar coordinates reduce the simultaneous control of all lower-left rectangular sums to an $\ell^\infty$ bound on the Haar coefficients, at a logarithmic price. Revealing the innovations one binary digit at a time writes these coefficients as a martingale with vanishing conditional third moments, so that the third-order Yurinskii coupling of Cattaneo et al. [ [14] applies without any symmetry assumption on the observations; this local coupling is the only source of the moment term $d/p$. A Gaussian-to-Gaussian comparison then replaces the resulting Gaussian blocks by an i.i.d Gaussian random field, while contributing as error the cost of dependence $(d-1)/2$. Finally, the Vorob’ev–Berkes–Philipp lemma stitches the local couplings into a single extension of the original probability space, on which the local errors are aggregated across dyadic scales.

Organization of the paper

Section 2 states the local approximation conditions, the strong approximation theorem, worked examples and the optimality result. Section 3 proves the main theorem. Appendices C and D collect auxiliary results and their proofs. Appendices A and B give the local Gaussian approximation and Gaussian-to-Gaussian comparison arguments.

Notation

We write $\mathbb{N}=\{1,2,\ldots\}$ and $[n]=\{1,\ldots,n\}$, and interpret inequalities between lattice points coordinatewise. For $m=(m_1,\ldots,m_d)\in\mathbb{N}^d$, define

Partial sums of other arrays are defined analogously. For a positive integer $n$, we use $m\leq n$ as shorthand for $m\in[n]^d$. For a finite set $Q$, $|Q|$ denotes its cardinality. For a vector $a\in\mathbb{R}^D$, $|a|$ denotes its Euclidean norm, and $|a|_\infty=\max_{1\leq j\leq D}|a_j|$. For $h\in\mathbb{Z}^d$ and a nonempty finite set $Q\subset\mathbb{Z}^d$, define

For a rectangle

we write $R \preceq Q$ if $R$ is a subrectangle anchored at the lower-left corner of $Q$, that is,

For a random vector $Z$ and $q\ge1$, let $\|Z\|_q=(\mathbb{E}|Z|^q)^{1/q}$ and $\|Z\|=\|Z\|_2$. For a matrix $M$, $\rho^*(M)$ denotes its largest singular value, which equals $\sup_{|a|=1}|a^\top M a|$ when $M$ is symmetric. For positive semidefinite $\Sigma$, $N(0,\Sigma)$ denotes the centered Gaussian law with covariance $\Sigma$, including the singular case. We write $i$ for the imaginary unit and $\mathcal{L}(Z)$ for the law of $Z$.

For a deterministic sequence $a_n>0$, we write $Z_n=O_{\mathrm{a.s.}}(a_n)$ if, almost surely, $|Z_n|\le C_0a_n$ for all $n\ge n_0$, where $C_0$ and $n_0$ are finite random variables. We use the usual deterministic Landau notation. We write $Z_n=O_{\mathrm{P}}(a_n)$ if $Z_n/a_n$ is bounded in probability and $Z_n=o_{\mathrm{P}}(a_n)$ if $Z_n/a_n$ converges to zero in probability. Finally, we write $a_n\lesssim b_n$ if $a_n\le Cb_n$ for some constant $C>0$, and $a_n\asymp b_n$ if both $a_n\lesssim b_n$ and $b_n\lesssim a_n$ hold.

Strong approximation and optimality

For $d\ge1$, we consider a centered stationary random field $(X_i)_{i\in\mathbb{Z}^d}$ with the Bernoulli representation

where $(\eta_j)_{j\in\mathbb{Z}^d}$ are independent and identically distributed real random variables on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, and $g:\mathbb{R}^{\mathbb{Z}^d}\to\mathbb{R}$ is measurable. In general, when $d\ge2$, due to the lack of spatial ordering, the above representation allows for dependence in all directions.

Following the functional-dependence framework introduced for time series by Wu [ [47] and developed for random fields by El Machkouri et al. [ [21], we measure dependence through the effect of changing a single innovation. Let $X_i^*$ be obtained from $X_i$ by replacing $\eta_0$ with an independent copy $\eta'_0$, while leaving all other innovations unchanged. For $q\ge1$, define