hold if the restriction $\Delta<\gamma/\sqrt{n}$ in the last display is replaced by $\Delta < g(n)$ provided that $\lim_{n\to\infty} g(n) = 0$. \[Here, the suprema are taken over all full-rank $d\times p$ matrices $M$, all $\vartheta\in {{\mathbb R}}$, all $d$-vectors $\theta$ and $\mu$, all distributions ${\mathcal L}(\epsilon)$ so that $\epsilon$ has mean zero and finite positive variance, and all symmetric and positive definite $d\times d$ matrices $\Sigma$, subject to the indicated restrictions.\]

\[pinsker\] Write ${\mathcal R}_N$ and ${\mathcal R}_L$ for the prediction risk of the Bayes predictor and of the best linear predictor, respectively, of $y$ given $x$. That is, ${\mathcal R}_N = {{\mathbb E}}[ (y - {{\mathbb E}}[y\|x])^2]$ and ${\mathcal R}_L = {{\mathbb E}}[ (y - (\alpha+\beta'x))^2]$. The results of @Ste18a then entail that, in the setting of Theorem \[t1\], ${\mathcal R}_N/{\mathcal R}_L$ converges to one, uniformly over all the parameters indicated in the last display of that theorem. In fact, the risk-ratio converges to one uniformly even if the restriction on $\Delta$ is removed altogether, and a similar statement holds for the ratio of conditional risks given $x$, i.e., $ {{\mathbb E}}[ (y - {{\mathbb E}}[y\|x])^2\|x]/{{\mathbb E}}[ (y - (\alpha+\beta'x))^2\|x]$. See Theorem 3.1 of @Ste18a for a more general 