= Jump chain of a discrete-time Markov chain
{title2=$\widetilde P_{ij}=P_{ij}/(1-P_{ii}),\quad i\ne j$}
= Discrete-time jump chain
{synonym}
For $P_{ii}<1$, observe a discrete-time <Markov chain> only when its state changes. The <probability> that its next different state is $j$ is $\sum_{r\geq0}P_{ii}^rP_{ij}=P_{ij}/(1-P_{ii})$. Its <holding times> are finite geometric waiting times. This construction deletes self-transitions and differs from the existing continuous-time jump-chain concept.
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