= Reflected lazy walk with doubled endpoint transitions
{title2=$\pi_0=\pi_m=1/(2m),\quad\pi_i=1/m\ (0<i<m)$}
On the finite path $\{0,\ldots,m\}$, take interior neighbor probabilities $p$, endpoint inward probability $2p$, and holding probability $1-2p$, with $0<p\leq1/2$. <Detailed balance> gives the displayed invariant distribution: endpoint weights are half the interior weights. For $p<1/2$ the chain is irreducible and aperiodic, so transition probabilities converge to it. At $p=1/2$ parity alternates and the period is two.
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