Stein Coupling Framework

• Stein Coupling Framework is a unifying structure that generalizes coupling techniques in Stein’s method to facilitate normal approximation.
• It provides explicit error bounds in Wasserstein and Kolmogorov metrics through transparent decomposition of remainder terms.
• Its versatility supports applications in combinatorial CLTs, occupancy schemes, and zero-bias enhancements for dependent random structures.

A Stein coupling framework provides a unifying structure for normal approximation within Stein's method, generalizing several coupling techniques—including exchangeable pairs and size biasing—by expressing the core normal approximation identity through a flexible triplet of random variables. Originating with Chen and Röllin (2010), Stein couplings facilitate quantitative bounds in the Wasserstein and Kolmogorov metrics, allow transparent decomposition of error terms, and have been enhanced by subsequent developments such as zero-bias enhancement. This framework subsumes numerous classical and modern approaches, leading to general and often sharp normal approximation results for a broad class of dependent structures in probability theory (Chen et al., 2010, Goldstein, 2022).

1. Core Definition and Fundamental Properties

Let $W$ be a real-valued random variable with $E[W^2]<\infty$. A Stein coupling is a triple $(W, W', G)$ of square-integrable real random variables such that, for every (say, Lipschitz) function $f$ for which the expectations exist,

$E[ G f(W') - G f(W) ] = E[ W f(W) ].$

Key consequences of this structural identity include:

• $E[W]=0$ (by testing $f\equiv 1$)
• With $D = W' - W$, $E[G D] = \operatorname{Var}(W)$ (by setting $f(x)=x$)
• Many classical couplings fall within this formulation, e.g., if $(W, W')$ is an exchangeable pair with $E[W'-W \mid W]=-\lambda W$, the triplet $(W, W', (W'-W)/(2\lambda))$ forms a Stein coupling.

This formulation unifies distinct coupling paradigms, notably the exchangeable-pair and size-bias couplings, as special cases (Chen et al., 2010, Goldstein, 2022).

2. Normal Approximation Theorems and Explicit Bounds

The main utility of the Stein coupling is in establishing explicit and general normal approximation bounds in the Wasserstein and Kolmogorov metrics. Error terms are decomposed into a finite palette of remainder terms, enabling routine analysis once a suitable Stein coupling is constructed.

Let $D = W' - W$ and, for an auxiliary random variable $S$ with $E[W S]=1$, define a sequence of error terms $r_0, r_1, \ldots, r_8$ that capture the various discrepancies arising in the coupling construction.

Wasserstein Bound (Theorem 2.1)

For any Stein coupling $(W, W', G)$ and auxiliary $S$, the Wasserstein distance between the law of $W$ and the standard normal satisfies

$$d_{\mathrm W}(\mathcal{L}(W), N(0,1)) \le 2 r_0 + 0.8 r_1 + 0.8 r_2 + 0.8 r_3 + 1.6 r_4 + 2 r_5 + 1.6 r_4 + 2 r_5.$$

If $(W, W', G)$ is an exact Stein coupling, $\operatorname{Var}(W)=1$, $S\equiv1$, and fourth moments are finite,

$$d_{\mathrm W}(\mathcal{L}(W), N(0,1)) \le 0.8 \operatorname{Var}(E[G D \mid W]) + E|G D|.$$

Kolmogorov Bound (Theorem 2.5)

With suitable truncation constants and under uniform bounds $|G| \le \alpha$, $|D| \le \beta$ and $S \equiv 1$, the Kolmogorov distance obeys

$$d_{\mathrm K}(\mathcal{L}(W), N(0,1)) \le 2\,\operatorname{Var}(E[G D \mid W]) + 8 \alpha \beta.$$

All constants and remainder terms are constructed to be both explicit and directly computable from the structure of the chosen coupling (Chen et al., 2010).

3. Methodological Unification and Generalizations

The Stein coupling framework subsumes and unifies previous methodologies in Stein's method:

• Exchangeable pair approach: If $(W, W')$ is an exchangeable pair with $E[W'-W|W]=-\lambda W$, defining $G=(W'-W)/(2\lambda)$ retrieves the classical regression-based coupling.
• Size-bias coupling: For $W \ge 0$ with mean $\mu$, the pair $(W^s, W, \mu)$, where $W^s$ is the $W$-size-biased version, is a Stein coupling (Chen et al., 2010).
• Zero-bias enhancement: Extending further, "zero-bias-enhanced Stein coupling" (zbest framework) allows for $(W'', W', G)$ (possibly on a different probability space), such that for all smooth $f$,

$E[(W-\mu)f(W)] = E_Q[G(f(W'')-f(W'))].$

This encompasses zero-bias couplings, with the key innovation that only first-moment (not conditional variance) information is necessary for sharp bounds (Goldstein, 2022).

• Approximate couplings: The framework accommodates approximate couplings, with any residual error increasing a single defect term.

This generality permits uniform analysis across a broad class of dependent and structured random variables, sidestepping the need for separate theoretical machinery in each context.

4. Proof Techniques and Error Control

The core proof mechanism leverages the solution $f$ to Stein's equation ($f'(w)-w f(w) = h(w) - E h(Z)$), enabling the normal approximation error for any real-valued $W$ to be converted into error terms involving $G$, $D$, and their conditional behaviors.

Highlighting the Wasserstein bound, the proof proceeds by:

• Expressing $E[h(W)] - E[h(Z)]$ in terms of $E[f'(W) - W f(W)]$
• Applying the Stein identity to relate $E[G f(W')-G f(W)-W f(W)] + r_0$
• Using the Fundamental Theorem of Calculus to expand $f(W')-f(W)$
• Carefully decomposing and bounding remainders using the terms $r_1, r_2, r_3, r_4, r_5$

In zbest generalizations, classical remainder terms involving variances of conditional expectations are replaced by simple first-moment controls, specifically $E|W^*-W|$, and a single defect term $E|1-G D|$ (Goldstein, 2022). This streamlines applications considerably compared to traditional exchangeable-pair or size-bias analysis.

5. Illustrative Applications

The versatility of Stein couplings is demonstrated via direct applications:

5.1 Hoeffding’s Combinatorial Central Limit Theorem

Consider random variables $\{a_{i,j}: 1 \le i,j \le n\}$ satisfying $\sum_k a_{i,k} = 0 = \sum_k a_{k,j}$ and $\sum_{i, j} a_{i, j}^2 = 1$. Sampling a random permutation $\pi$ of $[n]$, let $W = \sum_{i=1}^n a_{i, \pi(i)}$.

A nonstandard “local-symmetry” Stein coupling is constructed by selecting two independent uniform indices $I_1, I_2$ and defining: $$W' = W - a_{I_1, \pi(I_1)} - a_{I_2, \pi(I_2)} + a_{I_1, \pi(I_2)} + a_{I_2, \pi(I_1)},$$

$$G = n \left( a_{I_1, \pi(I_2)} - a_{I_1, \pi(I_1)} \right).$$

Applying the Kolmogorov bound yields for $a = \max_{i,j} |a_{i,j}|$,

$$d_{\mathrm K}(\mathcal{L}(W), N(0,1)) \le 448 n a^2 + 96 a^2.$$

This rate recovers the known $O(n^{-1/2})$ optimality but with transparent coupling and explicit constants (Chen et al., 2010).

5.2 Occupancy Scheme Functionals

Given $m$ balls placed independently in $n$ urns with probabilities $p_1,\dots,p_n$, and statistic $U = \sum_{i=1}^n h_i(\xi_i)$ (with $\xi_i$ the count in urn $i$), the coupling is defined by randomly deleting an urn and re-distributing its balls. Setting

$G = -\frac{n}{\sigma} (h_I(\xi_I) - E h_I(\xi_I))$

yields, under mild moment conditions,

$$d_{\mathrm K}(\mathcal{L}(W), N(0,1)) \le C \frac{n (\Delta h)^3 \ln(nh) (1+\sigma^2/n)}{\sigma^3} + \frac{3888}{\ln(nh)},$$

where $\Delta h = \max_i \sup_x |h_i(x+1)-h_i(x)|$, $h = \max_i \sup_x |h_i(x)|$. For uniform urns and suitable moment bounds, this recovers the optimal normal approximation rate (up to a logarithmic factor) (Chen et al., 2010).

5.3 Zero-Bias Enhancement: Lightbulb Process

In the lightbulb process, where $n$ bulbs are toggled through $n$ random stages, zbest couplings yield bounds for the standardized number $W$ of 'on' bulbs: $$d(W, Z) \le \frac{6}{\sigma}, \quad d_K(W, Z) \le \frac{8.12}{\sigma}$$ with $\sigma^2 = \operatorname{Var}(Y)$. These constants improve upon earlier results and require only first-moment controls (no conditional variances) (Goldstein, 2022).

6. Extensions: Zero-Bias and Approximate Couplings

Zero-bias enhancement (zbest) generalizes Stein couplings to a triplet $(W'', W', G)$ (possibly defined on an auxiliary measure $Q$), linked via

$E[(W-\mu)f(W)] = E_Q[G(f(W'') - f(W'))].$

Defining $D = W'' - W'$, $W^* = U W'' + (1-U) W'$, with $U \sim \operatorname{Unif}[0,1]$, the coupling yields

$$E[(W-\mu)f(W)] = E_Q[G D f'(W^*)], \quad E_Q[G D] = \sigma^2.$$

Normal approximation bounds in both Wasserstein and Kolmogorov metrics are then given solely in terms of the shift $E|W^*-W|$ and the defect $E|1-G D|$, without the need for higher-order conditional variance quantities. This unifies exchangeable-pair, size-bias, and zero-bias approaches and accommodates approximate couplings: any residual error is absorbed additively into the defect term (Goldstein, 2022).

7. Impact and Current Research Directions

The Stein coupling framework has systematically clarified, unified, and extended normal approximation results across a variety of probabilistic models, from classical combinatorial structures to modern dependency graphs and point processes. It has facilitated the plug-and-play construction of bounds with transparent error decomposition and explicit constants, and forms the foundation for recent progress in zero-bias enhancement and 'defect'-controlled approximate couplings. Current research leverages this flexibility for further generalizations, sharper constants, and streamlined proofs in increasingly high-dimensional and dependent settings (Chen et al., 2010, Goldstein, 2022).