On a contractivity question of Neri and Pinto for generalized Pólya urns

Let $R$ be the replacement matrix of a balanced $d$-colour generalized Pólya urn with balance $\lambda$. Neri and Pinto obtained quantitative rates in the strong law $U(n)/T(n)\to u$ whenever the quadratic form $w\mapsto\langle w(R-\lambda I),w\rangle$, restricted to vectors with coordinate sum zero, is negative definite for the Euclidean inner product on all $d$ coordinates or on the first $d-1$ of them, and they asked whether this holds for every tenable irreducible urn. We show that it holds for every irreducible urn with two colours and fails for some urn with nonnegative integer entries for every $d\ge 3$, simultaneously for both inner products and for every ordering of the colours. One of the examples has all non-principal eigenvalues of real part below $\lambda/2$. On the other hand, the Lyapunov equation on the sum-zero hyperplane gives, for every tenable irreducible urn, an inner product in which the form is negative definite, and the best constant obtainable in this way is $\lambda-\gamma$, where $\gamma$ is the largest real part of a non-principal eigenvalue of $R$. Consequently the general rate theorem of Neri and Pinto applies to every tenable irreducible urn, and their fast-rate theorem applies with such an inner product exactly when $\gamma<\lambda/2$.