= Gaussian autoregressive proposal reversible with respect to a standard normal distribution
Let $0<\beta\leq1$, let $X\in\mathbb R^p$, and propose
$$
Y=\sqrt{1-\beta^2}\,X+\beta Z,
\qquad Z\sim N(0,I_p).
$$
The <proposal distribution> is $N(\sqrt{1-\beta^2}\,X,\beta^2I_p)$. If $X\sim N(0,I_p)$ independently of $Z$, then $(X,Y)$ is a jointly <multivariate normal distribution> invariant under exchanging $X$ and $Y$, because both <random vectors> have <covariance matrix> $I_p$ and their cross-covariance matrices are both $\sqrt{1-\beta^2}I_p$. Consequently its density $q$ satisfies
$$
\phi(x)q(y\mid x)=\phi(y)q(x\mid y),
$$
where $\phi$ is the standard-normal density. Thus the proposal is <reversible Markov chain>[reversible] with respect to the standard normal distribution.
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