Given a posterior distribution $\pi(\theta|X)$, we would like it to support the real line $\mathbb{R}$ through MCMC(Markov chain Monte Carlo). For a given function,I would like to find out whether it is a valid transition function for a Markov Chain that converges to the posterior distribution(i.e,converges to $\pi(\theta|X)$). Let $\begin{equation} \theta^{(t)}= \begin{cases} \theta^{t-1}&\mbox{with prob = r}\\ \theta^{'}&\mbox{with prob = 1-r} \end{cases} \end{equation} $ Where $\theta^{'} \sim \theta^{t-1} + N(0,1)$
and $r = min(1,\frac{\pi(\theta^{(t-1)}|X)}{\pi(\theta^{'}|X)})$.
I have no idea where to start.Is there any property that this function need to satisfy to converge to the posterior? Thanks in advance.