= Poisson Metropolis chain
{c}
{title2=$u_n=\tfrac12\min(1,\lambda/(n+1)),\ d_n=\tfrac12\min(1,n/\lambda)$}
A <Poisson Metropolis chain> samples a <Poisson distribution> by proposing an upward or downward unit step with equal <probability> and reflecting a downward proposal at zero into a self-proposal. Its upward and downward <transition probabilities> are the displayed $u_n,d_n$, with $d_0=0$, and the diagonal entry is $1-u_n-d_n$. For $\pi_n=e^{-\lambda}\lambda^n/n!$, adjacent stationary fluxes equal $\tfrac12\min(\pi_n,\pi_{n+1})$. Hence <detailed balance> holds. Positive adjacent <probabilities> imply an <irreducible Markov chain>; $P(0,0)\geq1/2$ implies <aperiodicity>. The normalized invariant <Poisson distribution> implies <positive recurrence>, so the <countable-state Markov chain convergence theorem> applies. The boundary proposal <probability> is essential: forcing every proposal at zero to go upward would require a different acceptance ratio.
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