= Poisson-sampled continuous-time Markov chain
{title2=$P_\lambda=\lambda(\lambda I-Q)^{-1}$}
Sample a finite-state <continuous-time Markov chain> with generator $Q$ at the independent arrival times of a rate-$\lambda$ <Poisson process>. The sampled states form a discrete-time Markov chain with transition matrix
$$
P_\lambda
=\int_0^\infty\lambda e^{-\lambda t}e^{tQ}\,dt
=\lambda(\lambda I-Q)^{-1}.
$$
Its invariant distributions are exactly those of the continuous-time chain.
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