Infinite-SNR asymptotic MMSE limit

Let $D(R,\beta)$ denote the asymptotic MMSE at measurement rate $R$ and inverse noise variance (SNR) parameter $\beta$, and let $m_a$ be the a-priori magnetization. In the low-noise regime, consider $\beta\to\infty$. Infinite SNR limit. For $R>m_a$, the asymptotic MMSE satisfies

$$\lim_{\beta\to\infty}\left[\beta\cdot D(R,\beta)\right]=\sigma^2\frac{m_a}{R-m_a}.$$

This conjecture describes the noise sensitivity in the high-SNR regime when the measurement rate exceeds the a-priori magnetization. The preceding discussion indicates that the asymptotic MMSE is governed by the first term in the MMSE expression in this range; the corresponding behavior for $R<m_a$ is that $\beta\cdot D(R,\beta)$ grows without bound, consistent with earlier observations for independent and identically distributed sources.