Gaussian Fluctuations Around Interior Targets

From Convergence to a Fluctuation Law

Consistency of targeted allocation proportions says that the allocation approaches its target. It does not describe the typical size or shape of the remaining error. A Gaussian fluctuation result answers a more refined question: after multiplying the error by \(\sqrt{n}\), does it approach a normal distribution, and what determines that distribution’s covariance?

The answer depends on more than the fact that the target is stable. The step size must be in the appropriate regime, the linearized drift must pull sufficiently strongly toward the target, and the random allocation or response noise must have a well-behaved limiting covariance. These conditions distinguish a root-\(n\) Gaussian limit from mere convergence.

Definition: Let \(X_n\) be a \(d\)-dimensional state representing allocation proportions in unconstrained local coordinates, let \(x^\ast\) be an interior target, and write \(e_n=X_n-x^\ast\). A root-\(n\) Gaussian fluctuation limit is a distributional convergence \(\sqrt{n}(X_n-x^\ast)\Rightarrow N(0,\Sigma)\), for a finite covariance matrix \(\Sigma\). If proportions satisfy a sum constraint, use coordinates on the constraint’s tangent space; the covariance is then expressed in those coordinates.

The recursion below is a local stochastic-approximation representation. Its noise may arise from randomized assignments, observed responses, or both. The theorem concerns the allocation state itself; it does not assert that every adaptive design has the same drift or noise covariance.

Assumptions for the root-\(n\) limit: In a neighborhood of \(x^\ast\), suppose \(X_n=X_{n-1}+\frac{c}{n}\{h(X_{n-1})+\xi_n\}+r_n\), where \(c>0\), \(h(x^\ast)=0\), and \(r_n\) is a remainder negligible at the root-\(n\) scale. Assume:

\(x^\ast\) is interior, the process converges to it, and the recursion remains in a region where the local expansion applies.
\(h\) is differentiable at \(x^\ast\), with Jacobian \(J\), and \(h(x^\ast+e)=Je+o(\|e\|)\).
\(\xi_n\) is a martingale-difference noise sequence: its conditional mean given the past is zero. Its conditional covariance converges to a deterministic matrix \(\Gamma\), and a conditional Lindeberg condition prevents a non-negligible contribution from rare large increments.
The accumulated effect of the nonlinear remainder and \(r_n\), after root-\(n\) scaling, vanishes in probability.
Every eigenvalue of \(A=cJ\) has real part less than \(-1/2\).

The final condition is stronger than the stability condition needed for convergence. It says that the local contraction is fast enough to overcome the amplification caused by multiplying the error by \(\sqrt{n}\). The covariance \(\Gamma\) describes the limiting conditional variability of \(\xi_n\), not the variability of \(X_n\) itself.

Why the Covariance Has a Shifted Lyapunov Equation

Set \(A=cJ\). Ignoring terms that are negligible at the root-\(n\) scale, linearization gives \(e_n\approx (I+A/n)e_{n-1}+(c/n)\xi_n\). Define \(Z_n=\sqrt{n}e_n\). Since \(\sqrt{n/(n-1)}=1+1/(2n)+O(n^{-2})\), the scaled recursion becomes

Z_n \approx \left[I+\frac{A+\tfrac12 I}{n}\right]Z_{n-1} +\frac{c}{\sqrt{n}}\xi_n.

The additional \(\tfrac12 I\) is essential: it comes from the changing scale \(\sqrt{n}\). The martingale central limit argument gives a Gaussian limit when the scaled drift is stable and the conditional noise variance settles to \(c^2\Gamma\). More explicitly, the covariance can be written as an integral of the accumulated noise:

\Sigma=\int_0^\infty \exp\!\left[(A+\tfrac12 I)t\right]c^2\Gamma \exp\!\left[(A^{\mathsf T}+\tfrac12 I)t\right]\,dt.

The eigenvalue condition makes the exponential terms decay, so the integral is finite. Differentiating the integrand with respect to \(t\), then integrating from zero to infinity, yields the covariance equation. This also explains why the equation has a half-identity shift rather than just the unshifted drift.

(A+\tfrac12 I)\Sigma+\Sigma(A^{\mathsf T}+\tfrac12 I)+c^2\Gamma=0.

Theorem: Under the stated local expansion, convergence, noise, remainder, and stability assumptions, \(\sqrt{n}(X_n-x^\ast)\Rightarrow N(0,\Sigma)\), where \(\Sigma\) is the unique solution of the displayed covariance equation. The eigenvalue condition \(\operatorname{Re}\lambda(A)<-1/2\) ensures the integral representation converges and gives a positive-semidefinite covariance when \(\Gamma\) is positive semidefinite.

Worked Example: A Scalar Allocation Proportion

Consider a hypothetical two-arm design, with \(X_n\) the proportion assigned to arm 1 and an interior target \(x^\ast=0.40\). Suppose its local recursion has \(c=1\), drift derivative \(J=-1\), and martingale noise with limiting conditional variance \(\Gamma=0.25\). These are assumed features of this hypothetical design, not estimates from a real trial.

Here \(A=cJ=-1\), so its eigenvalue is \(-1<-1/2\), satisfying the root-\(n\) stability condition. In one dimension the covariance equation is \((A+\tfrac12)\Sigma+\Sigma(A+\tfrac12)+c^2\Gamma=0\). Substitution gives \[ (-1+0.5)\Sigma+\Sigma(-1+0.5)+0.25=0, \] so \(-\Sigma+0.25=0\), and \(\Sigma=0.25\).

Thus \(\sqrt{n}(X_n-0.40)\) converges in distribution to \(N(0,0.25)\). For \(n=400\), the corresponding first-order standard deviation of \(X_n\) is \(\sqrt{0.25/400}=0.0250\), in proportion units. This is an asymptotic approximation, not an exact finite-sample standard deviation.

Worked Example: A More Weakly Contracting Target

For a second hypothetical scalar allocation process, suppose the target is \(0.35\), \(c=1\), \(J=-0.8\), and the limiting conditional noise variance is \(0.36\). The target is interior, and the local drift is restoring because \(J<0\). Check, however, whether the stronger root-\(n\) condition holds.

The eigenvalue is \(A=-0.8\), which is less than \(-0.5\), so the condition does hold. The scalar covariance equation becomes \[ 2(-0.8+0.5)\Sigma+0.36=0. \] Therefore \(-0.6\Sigma+0.36=0\), giving \(\Sigma=0.60\). The limiting distribution is \(N(0,0.60)\) for \(\sqrt{n}(X_n-0.35)\). At \(n=900\), the approximate standard deviation of the unscaled proportion is \(\sqrt{0.60/900}=0.0258\).

This example separates the stability check from the covariance calculation: a negative drift derivative alone is not enough. Its magnitude must also meet the root-\(n\) threshold.

Worked Example: A Two-Coordinate Covariance

Suppose a hypothetical interior allocation target is represented by two local coordinates. Assume \(c=1\), \(A=J=\begin{pmatrix}-1&0\\0&-2\end{pmatrix}\), and limiting noise covariance \(\Gamma=\begin{pmatrix}0.4&0.1\\0.1&0.8\end{pmatrix}\). This noise covariance is valid: its diagonal entries are positive and its determinant is \(0.4(0.8)-0.1^2=0.31>0\).

Both eigenvalues of \(A\) are below \(-1/2\). Write \(\Sigma=\begin{pmatrix}u&v\\v&w\end{pmatrix}\). The first diagonal covariance equation is \(2(-1+0.5)u+0.4=0\), so \(u=0.4\). The second is \(2(-2+0.5)w+0.8=0\), so \(w=0.2667\). The off-diagonal equation is \[ (-1+0.5)v+v(-2+0.5)+0.1=0, \] or \(-2v+0.1=0\), giving \(v=0.05\). Thus \[ \Sigma=\begin{pmatrix}0.4&0.05\\0.05&0.2667\end{pmatrix}. \]

The positive off-diagonal entry reflects the assumed positive covariance between the two noise coordinates. The matrix is positive definite, since its determinant is approximately \(0.1042>0\). The result describes joint root-\(n\) fluctuations in the chosen coordinates.

Step Size, Diagnostics, and Alternatives

For the gain \(c/n\), a useful diagnostic is to calculate the eigenvalues of \(cJ\) and compare their real parts with \(-1/2\). If any real part is at least \(-1/2\), the stated root-\(n\) theorem does not apply. This does not by itself prove that the process fails to converge; it means the proposed normalization and covariance formula have not been justified by these conditions.

The step-size regime matters. With gains proportional to \(n^{-\rho}\), where \(1/2<\rho<1\), fluctuations are generally considered on a scale related to the reciprocal square root of the gain, rather than automatically on the \(\sqrt{n}\) scale. The \(c/n\) recursion has a special threshold because drift and noise accumulate over the same logarithmic time scale after root-\(n\) rescaling. Use the gain actually present in the allocation recursion; do not substitute a familiar formula from a different regime.

Several checks can reveal a mismatch between a design and the theorem’s assumptions:

Estimate or derive the local Jacobian and verify that the target is genuinely locally attracting, with the stronger root-\(n\) eigenvalue condition.
Check that the conditional noise covariance approaches a fixed finite matrix. Persistent changes in response variability or allocation feedback may invalidate a deterministic \(\Gamma\).
Check whether occasional large increments violate the conditional Lindeberg requirement. A covariance calculation alone does not rule out influential rare jumps.
Confirm that the target is interior and that the local coordinates remain appropriate. Near a boundary, the assumptions and scaling may change.

The main alternative is to report only consistency, which describes the limiting target but not the scale or shape of the error. Another tempting alternative is to apply an ordinary independent-and-identically-distributed central limit theorem directly to assignments; adaptive assignments are generally dependent through their history. The martingale central limit framework addresses that dependence, while the linearized recursion adds the allocation feedback that determines \(\Sigma\). If conditional noise variance has a nonconstant random limit, a fixed covariance Gaussian theorem needs modification; the deterministic-\(\Gamma\) result above should not be used without checking that assumption.

Common Mistakes and Limits

Using stability as if it guaranteed a root-\(n\) limit. A negative drift eigenvalue can support convergence while failing the stricter requirement \(\operatorname{Re}\lambda(cJ)<-1/2\).
Omitting the half-identity shift. The \(\tfrac12 I\) term arises when scaling \(e_n\) by \(\sqrt{n}\); using an unshifted Lyapunov equation generally gives the wrong covariance.
Treating adaptive increments as independent. Their conditional distributions depend on the past. The relevant noise condition is a martingale-difference and conditional-variance condition, not an assumption of independent assignments.
Interpreting \(\Sigma\) as the covariance of \(X_n\) at every sample size. It is the limiting covariance of the scaled error. The approximation \(\operatorname{Cov}(X_n)\approx\Sigma/n\) is asymptotic and may be poor at small \(n\).

This result is local and conditional on its assumptions. It does not establish that a particular allocation rule has the required drift, that its target is reached from every starting point, or that finite-sample fluctuations are Gaussian. Those properties require separate analysis of the design and its operating region.

Check Your Understanding

Answer each question, then compare with the solutions.

For a scalar recursion with \(c=1\) and \(J=-0.4\), does the root-\(n\) stability condition hold? What can you conclude?
With \(c=1\), \(J=-1.5\), and \(\Gamma=0.6\), calculate the scalar limiting variance.
Why does the covariance equation include \(\tfrac12 I\)?
What noise condition is needed in place of treating adaptive assignments as independent observations?

Solutions. (1) No: \(-0.4\) is not less than \(-1/2\). The stated root-\(n\) theorem does not apply; this alone does not rule out convergence. (2) The equation is \(2(-1.5+0.5)\Sigma+0.6=0\), so \(-2\Sigma+0.6=0\) and \(\Sigma=0.30\). (3) It comes from \(\sqrt{n/(n-1)}=1+1/(2n)+O(n^{-2})\) when the error recursion is rescaled by \(\sqrt{n}\). (4) Require martingale-difference noise with conditional covariance converging to the specified limit and a conditional Lindeberg condition.

Gaussian Fluctuations Around Interior Targets

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