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  "body": "\n## Definition\n\nLet $(X_n)$ be a time-homogeneous [[Markov chain|033TJBZDG0]] with [[transition matrix|0F1KBVY161]] $P = (p_{ij})$ on a state space $S$.\nA state $i \\in S$ is absorbing if $p_{ii} = 1$. A state that is not absorbing is transient.\nThe chain is an absorbing Markov chain if it has at least one absorbing state and each state can reach an absorbing state.\nThat is, for each $i \\in S$ there are an absorbing state $j$ and an $n \\ge 0$ with $p^{(n)}_{ij} \u003e 0$.\n\n## Remarks\n\nFor a finite absorbing chain, write $P$ in the canonical form with the transient states first: $P = \\begin{pmatrix} Q \u0026 R \\\\ 0 \u0026 I \\end{pmatrix}$.\nThe fundamental matrix is $N = (I - Q)^{-1} = \\sum_{n=0}^{\\infty} Q^n$. The entry $N_{ij}$ is the expected number of visits to the transient state $j$, starting from the transient state $i$.\nThe expected number of steps before absorption from $i$ is the $i$-th entry of $N \\mathbf{1}$, where $\\mathbf{1}$ is the column vector of ones. The absorption probabilities are the entries of $NR$.\nA finite absorbing chain reaches an absorbing state with probability $1$.\n\n## Explanations\n\n### L1 human\n\nAn absorbing chain wanders for a while and then gets stuck in a trap state forever.\nThe fortune in the gambler's ruin problem is an example: the states $0$ and $N$ are traps.\n",
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    "name": "Absorbing Markov chain",
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  "name": "Absorbing Markov chain",
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