Stein's Method with Iterative State-Space Peeling

• The paper introduces a framework that decomposes approximation error iteratively through symmetric interpolation and Stein coupling to manage dependencies.
• The methodology provides explicit error bounds by controlling local and global dependencies between complex random vectors and their Gaussian counterparts.
• The approach is applied to complex models like spin glasses and percolation, offering rigorous insights into universality and layered state-space analysis.

Stein’s method with iterative state-space peeling is a framework for quantifying the difference between the law of a high-dimensional, potentially dependent random vector and that of a Gaussian reference, by symmetrically decomposing the error and systematically controlling dependencies through coupling constructions. The technique generalizes classical central limit theorem arguments to high-dimensional, locally or globally dependent models, and provides a blueprint for decomposing distributional error in iterative, layer-wise fashion for complex state spaces.

1. Symmetric Interpolation and Error Decomposition in High Dimensions

The approach begins by interpolating between the random vector $X=(X_1,\ldots,X_n)$ (not assumed independent) and a Gaussian vector $Z$ with matched covariance. The interpolation is performed via the path

$Y_t = V_t X + V_{1-t} Z,$

with $V_t = t^{1/2} I$, $0 < t \leq 1$. This construction treats all coordinates symmetrically, avoiding the sequential bias of the Lindeberg telescoping sum.

For any three-times partially differentiable function $h:\mathbb{R}^n\to\mathbb{R}$, the difference in expectation can be written using the partial derivatives $h_i$:

$$\mathbb{E}[h(X)] - \mathbb{E}[h(Z)] = \frac{1}{2} \int_0^1 \left\{ \frac{1}{t} \sum_{i=1}^n \mathbb{E}[X_i h_i(Y_t)] - \frac{1}{1-t} \sum_{i=1}^n \mathbb{E}[Z_i h_i(Y_t)] \right\} dt$$

(see equation (2.3)). This formula provides the foundation for decomposing the error in a permutation-invariant, symmetric fashion.

A key refinement—Lemma 2.1 in (Röllin, 2011)—expresses the difference as a sum of three terms,

$$\mathbb{E}[h(X)] - \mathbb{E}[h(Z)] = \int_0^1 R_1(t)\,dt + \iint R_2(t,s)\,ds dt + \iint R_3(t,s)\,ds dt,$$

yielding the bound

$$|\mathbb{E}[h(X)] - \mathbb{E}[h(Z)]| \leq \sup_t |\mathbb{E}[R_1(t)]| + \sup_{t,s} |\mathbb{E}[R_2(t,s)]| + \sup_{t,s} |\mathbb{E}[R_3(t,s)]|.$$

Each $R_i$ term quantifies a particular layer of approximation error, involving covariance mismatches and higher-order corrections that isolate the contribution of local dependence and nonlinearities.

2. Stein Coupling Framework and Local Dependence Management

Stein couplings are central to handling dependencies within $X$. A Stein coupling is a triple $(X, X', G)$ such that, for smooth $f$,

$$\mathbb{E}[X_i f_i(X)] = \mathbb{E}[G_i (f_i(X') - f_i(X))]$$

(Definition 2.1). For independent $X$, $X'$ is typically constructed by resampling a coordinate, and $G$ is proportional to the difference. Under local dependence—where the influence of $X_i$ is restricted to a neighborhood—the construction is adapted to ensure the above relation holds approximately.

Error bounds (Lemma 2.2) depend on moments such as $E|G D|$ and $E|G| E|D|^2$, where $D=X'-X$. These bounds are tight when the coordinate-wise change ($D$) and the coupling ($G$) are small or localized, making the framework well-suited for models with local interactions.

3. Applications to Statistical Mechanics and Percolation Models

Sherrington–Kirkpatrick Spin Glass Model

The SK model assigns a Hamiltonian

$$H_N(\sigma) = \sqrt{\frac{2}{N}} \sum_{i,j} S_{ij}\,\sigma_i \sigma_j,$$

on spin configurations $\sigma$. Even with dependent $S_{ij}$ (environment), Stein's method controls the error between the true log-partition function and its Gaussianized analogue:

$$|\mathbb{E} \log Z_N(\beta, S) - \mathbb{E} \log Z_N(\beta, g)|,$$

where $g$ is Gaussian with matched covariance. The error bounds, via Stein couplings and Lemma 4.1/Theorem 4.2, establish universality of the Parisi formula for the free energy in environments with limited local dependence.

Last Passage Percolation on Thin Rectangles

For lattice paths, the non-differentiable maximum is replaced by a smooth softmax surrogate:

$$f_\epsilon(x) = \mathbb{E}[\log(\sum_p \exp(y_p))],$$

with upper bound $0 \leq f_\epsilon(x) - f_0(x) \leq \log(m)$. The main theorem (4.5) provides explicit error bounds for comparing functionals of the percolation process to their Gaussian counterparts:

$$|\mathbb{E} g(P_x) - \mathbb{E} g(P_z)| < C n^{1/3} k^{7/6} (\log m)^{2/3}$$

for $g$ three-times differentiable, where $n$ is the local dependency size and $k$ the rectangle thinness.

4. Conceptual Parallel: Iterative State-Space Peeling

Though not named in the paper, the technique shares a key philosophy with iterative state-space peeling. The interpolating path $Y_t$ naturally decomposes the error into contributions associated with progressive transitions from $X$ to $Z$, with each term $R_i$ representing an unreconciled layer of dependency.

In models with local structure (Curie-Weiss, SK, percolation), the Stein coupling construction allows one to control and "peel off" the effect of neighborhoods or blocks iteratively. This suggests a strategy for reducing complex high-dimensional problems into manageable layers, each associated with local corrections quantified by higher-order derivatives and coupling bounds.

The analytic framework resembles an iterative algorithm: at each stage, one peels off a level of dependency, controls the residual via Stein-type error bounds, and aggregates these to obtain global distributional approximations.

5. Summary Table: Central Technical Ingredients

Ingredient	Role	Quantitative Bound Examples
Symmetric Interpolation	Decomposes error globally	Formulas (2.3), (2.4), integrating over $t,s$
Stein Coupling	Manages local/global dependencies	$E[X_i f_i(X)] = E[G_i(f_i(X') - f_i(X))]$
Error Terms ($R_1, R_2, R_3$)	Layered error decomposition	$|E[h(X)] - E[h(Z)]| \leq \sup E|R_1|+\cdots$
Application-specific constructions	Adapt method to SK, percolation, occupancy	Error bounds depend on local dependence structure and function smoothness

6. Broader Significance and Implementation Implications

Stein’s method with iterative state-space peeling generalizes invariance principles and central limit results to complex, high-dimensional random vectors with potentially strong or local dependencies. By systematizing the error decomposition via symmetric interpolation and coupling constructions, it enables both explicit analytic bounds and computational blueprints for probabilistic approximations in settings such as spin glasses, percolation, occupancy problems, and beyond.

The separation of error into coordinate/local contributions aligns closely with algorithms that aim to incrementally reconcile state-space dependence, for example, in high-dimensional statistical physics, combinatorial optimization, or probabilistic graphical models. The framework thus supports both theoretical analysis of universality and quantitative performance bounds for approximation schemes operating in layered or blockwise fashion.

In summary, the iterative decomposition inherent to this variant of Stein’s method provides rigorous pathways for understanding and controlling the propagation of dependency-induced error across successive "peels" of the high-dimensional state space. This underpins both sharp error analysis and algorithmic design in a variety of complex random systems.