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  "body": "\n## Statement\n\nLet $P$ be an $n \\times n$ [[transition matrix|0F462493EA]] of a finite [[Markov chain|033TJBZDG0]] with states $S_1, \\dots, S_n$.\nCall $P$ regular if some power $P^k$ has only positive entries.\n\nAssume that $P$ is regular. Then there is a row vector $W = (w_1, \\dots, w_n)$ with these properties:\n\n1. every $w_i$ is a positive real number, and $w_1 + \\dots + w_n = 1$;\n2. as $m \\to \\infty$, every row of $P^m$ converges to $W$;\n3. $WP = W$, so $W$ is a [[stationary distribution|0KRH5157WP]];\n4. for each $i$, the number $w_i$ is the limiting probability that the chain is in state $S_i$, whatever the starting state.\n\n## Remarks\n\nThe vector $W$ is unique. It is the only probability vector fixed by $P$.\n",
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  "name": "Limiting distribution of a regular transition matrix",
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