A point-mass Gaussian prior excluding the truth prevents contraction

The prior ${\delta }_0=N(0,0)$ is a proper centered Gaussian law. Under its prior-predictive joint law, $\theta =0$ almost surely, so for every Borel $A\subseteq {\mathrm{\ell }}^2$ and product-Borel $B\subseteq {\mathbb{R}}^{{\mathbb{N}}_{>0}}$, $\mathbb{P}(\theta \in A,Y\in B)={\mathbf{1}}_A(0){\mathbb{P}}_0^{ϵ}(Y\in B)={\int }_B{\delta }_0(A){\mathbb{P}}_0^{ϵ}(dy)$. Thus the constant kernel $y\mapsto {\delta }_0$ is a posterior version. The prior assigns zero mass to every ball centered at $e_1$ of radius less than $1$.

Under the fixed truth $e_1$, only the first data coordinate shifts. Relative to the prior-predictive data law, the one-coordinate normal density ratio is $L(y)=\exp \text{\hspace{0.17em}⁣}(\frac{s_1{y}_1}{{ϵ}^2}-\frac{s_1^2}{2{ϵ}^2})$, and it integrates to $1$ under the prior-predictive first-coordinate law. It is positive and finite for every $y$ by [F4]–[F6]. All remaining coordinates have identical independent laws. The two full product-Borel data laws therefore agree as finite measures on the generating cylinder $\pi$-system after weighting by $L$; [F5] extends the identity to all data events. Since $L>0$, the laws are equivalent, so the posterior version ${\delta }_0$ remains the same almost surely under the truth law.

For any candidate rate $r_{ϵ}\to 0$, take $M_{ϵ}=r_{ϵ}^{-1\mathrm{/}2}$ for sufficiently small $ϵ$. Then $M_{ϵ}\to \mathrm{\infty }$ and $M_{ϵ}{r}_{ϵ}=\sqrt{r_{ϵ}}<1$ eventually. The posterior is concentrated at $0$, whose distance from $e_1$ is $1$, so its mass outside the radius-$M_{ϵ}{r}_{ϵ}$ ball is exactly $1$ under the truth law. This violates [F7] and refutes the Statement. The example lies outside the positive-coordinate-variance hypotheses in [F2].