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  "body": "\n## Definition\n\nLet $P = (p_{ij})_{i, j \\in S}$ be the [[transition matrix|0F1KBVY161]] of a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$.\nA stationary distribution of the chain is a family $\\pi = (\\pi_j)_{j \\in S}$ of real numbers with $\\pi_j \\ge 0$ for each $j$, $\\sum_{j \\in S} \\pi_j = 1$, and\n$$\\pi_j = \\sum_{i \\in S} \\pi_i \\, p_{ij} \\quad \\text{for each } j \\in S.$$\nIn matrix form, $\\pi P = \\pi$ for the row vector $\\pi$.\n\n## Remarks\n\nIf $X_0$ has the law $\\pi$, then $X_n$ has the law $\\pi$ for each $n \\ge 0$.\nAn irreducible chain on a finite state space has exactly one stationary distribution. Its entries are positive, and $\\pi_j = 1 / E_j[T_j]$, where $T_j$ is the time of the first return to $j$.\nFor an irreducible and aperiodic finite chain, $P(X_n = j) \\to \\pi_j$ as $n \\to \\infty$, for each initial law.\nThe simple random walk on $\\mathbb{Z}$ has no stationary distribution.\n\n## Explanations\n\n### L1 human\n\nA stationary distribution is a balance: the flow of probability into each state equals the flow out.\nFor many chains, it also describes the long-run fraction of time that the chain spends in each state.\n",
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