Dominant Eigenvalues and Non-Gaussian Urn Limits

Why a dominant nonleading eigenvalue changes the limit

In a balanced urn, the leading eigenvalue determines the long-run allocation proportions. The other eigenvalues describe how deviations from those proportions evolve. When the largest nonleading eigenvalue is below half the balance rate, those deviations typically fluctuate at the square-root rate and have a Gaussian limit. At the threshold, logarithmic normalization enters. Above the threshold, one nonleading direction decays more slowly than ordinary sampling noise: its limit is random and is generally not Gaussian.

The key consequence is a change in both rate and distribution. If the balance rate is \(s\) and the dominant nonleading eigenvalue is \(\lambda>s/2\), the corresponding proportion deviation is of order \(n^{-(1-\lambda/s)}\). Since \(1-\lambda/s<1/2\), it shrinks more slowly than \(n^{-1/2}\). The limit law depends on the urn’s dynamics; there is no universal normal approximation for this component.

Model and assumptions

Let \(Y_n\) be a column vector of counts or masses of \(d\) colors after \(n\) draws, and let \(T_n=\mathbf{1}^{T}Y_n\) be the total mass. At draw \(n+1\), color \(j\) is selected with probability \(Y_{n,j}/T_n\). The urn then receives a replacement vector \(\xi_{n+1}\). Define the replacement-mean matrix \(A\) by setting its \(j\)th column to the conditional mean replacement vector when color \(j\) is drawn.

For the result below, assume the urn is balanced: the total added mass is \(s>0\) at every draw, so \(T_n=T_0+ns\). Thus every column of \(A\) sums to \(s\). Assume \(A\) is nonnegative and primitive, so its leading eigenvalue \(s\) is simple and has a positive right eigenvector \(v\), normalized by \(\mathbf{1}^{T}v=1\). Assume the replacement vectors are bounded (a suitable finite-moment condition can replace boundedness).

In addition, assume \(A\) has a real, simple, nonleading eigenvalue \(\lambda\) such that \(\lambda>s/2\). Let \(r\) and \(w^{T}\) be corresponding right and left eigenvectors, normalized so that \(w^{T}r=1\). Because \(\lambda\ne s\), \(w^{T}v=0\). Assume every other nonleading eigenvalue has real part strictly less than \(\lambda\). These conditions make the \(\lambda\)-direction the unique slowest-decaying direction away from the target.

The limit and why its rate is nonstandard

Write \(Z_n=Y_n/T_n\) for the vector of urn proportions. Under the assumptions above, the dominant-eigenvalue limit has the form

where convergence is in an appropriate probabilistic sense (in particular, the projected limit can be obtained in \(L^2\) under the bounded-replacement assumptions). The scalar \(W\) is a random limit determined by the initial urn and its subsequent draws and replacements. It may be degenerate for special urns; when nondegenerate, its distribution is generally model-specific and need not be Gaussian. The result does not assert that every urn in this regime has a non-Gaussian limit.

A short martingale argument explains the normalization. Set \(X_n=w^{T}Y_n\). Conditional on the current urn, the expected replacement is \(AY_n/T_n\). Since \(w^{T}A=\lambda w^{T}\),

Define \(P_0=1\) and \(P_n=\prod_{k=0}^{n-1}(1+\lambda/T_k)\). Then \(X_n/P_n\) is a martingale. Since \(T_k=T_0+ks\), the product satisfies \(P_n\sim Cn^{\lambda/s}\) for a positive constant \(C\). The martingale’s bounded-increment differences, after division by \(P_n\), have variances bounded by a constant times \(n^{-2\lambda/s}\). The series of these bounds converges because \(2\lambda/s>1\). Thus the normalized projection has a finite random limit, and \(n^{-\lambda/s}X_n\) has a finite random limit as well.

Because \(w^{T}v=0\), the projection of the proportion deviation is \(w^{T}(Z_n-v)=X_n/T_n\). Also \(T_n/n\to s\), so

This converges to a random scalar. The other eigendirections vanish under this normalization: their eigenvalues have smaller real parts, and ordinary noise of order \(n^{-1/2}\) also vanishes after multiplication by \(n^{1-\lambda/s}\), since \(1-\lambda/s<1/2\). Together these facts give the stated vector limit in the dominant direction.

Reading the rate in a realized urn path

The theorem concerns a limit over repeated draws, not a claim that every short path follows a smooth curve. A finite path is useful for checking how the eigenvector direction appears, but it cannot establish the limiting distribution.

Diagnostics, alternatives, and limitations

Before applying the result, check the model structure rather than inferring the regime from a noisy plot. Verify that total mass added per draw is constant, that the replacement matrix uses the stated column convention, and that its leading eigenvalue and normalized target are correct. Then inspect the nonleading eigenvalues: the dominant condition is a real simple \(\lambda>s/2\) that exceeds the real parts of the other nonleading eigenvalues. If eigenvalues are close, numerical eigenvalue estimates may not reliably distinguish the regimes.

A useful diagnostic is to examine the projected contrast \(w^TY_n\) across simulated urn paths or repeated hypothetical runs. In the dominant regime, plotting \(n^{-\lambda/s}w^TY_n\) should be more informative about stabilization than plotting a square-root-scaled proportion contrast. Persistent drift under this scaling can signal an incorrect eigenvalue, a violated balance assumption, or a competing eigenvalue not accounted for. A single path cannot verify a distributional limit; diagnostics should be interpreted alongside the model assumptions.

The main alternatives are distinguished by the eigenvalue threshold. If the largest nonleading real part is below \(s/2\), ordinary sampling noise can dominate and a square-root Gaussian limit is typical. At \(s/2\), the critical regime requires logarithmic normalization. Above \(s/2\), the eigenvalue-driven random term dominates, with the slower rate derived here. These regimes cannot be interchanged by merely changing a standard error formula.

The theorem also has limits. It relies on balance, regularity of the replacement process, and a separated dominant eigenvalue. Random total additions, unbounded replacements without adequate moment control, a repeated or complex dominant eigenvalue, or a tie between nonleading eigenvalues can require different arguments and possibly different limiting forms. Further, \(W\) is not supplied by the eigenvalue calculation alone: its law depends on the full urn mechanism and initial state.

Common mistake

Wrong reasoning: “The urn proportions converge to \(v\), so their errors must be of order \(n^{-1/2}\) and approximately normal.” Consistency only says the errors approach zero; it does not determine their rate or limiting distribution.

Correction: Compare the nonleading eigenvalues with \(s/2\). When a simple dominant eigenvalue satisfies \(\lambda>s/2\), use the rate \(n^{1-\lambda/s}\) in its direction and expect a model-specific random coefficient. Do not report a conventional square-root Gaussian approximation for that component without additional justification.