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  "body": "\n## Statement\n\nLet $(X_n)_{n \\ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable set $S$ with [[transition matrix|0F462493EA]] $P = (p_{ij})$, and write $p^{(n)}_{ij}$ for the entries of $P^n$.\nAssume that:\n\n1. the chain is irreducible: for all $i, j \\in S$ there is $n \\ge 0$ with $p^{(n)}_{ij} \u003e 0$;\n2. the chain is aperiodic: for each $i \\in S$, the greatest common divisor of $\\set{n \\ge 1 : p^{(n)}_{ii} \u003e 0}$ is $1$;\n3. the chain has a [[stationary distribution|0KRH5157WP]] $\\pi$.\n\nThen, for every law of $X_0$,\n$$\\lim_{n \\to \\infty} \\sum_{j \\in S} \\abs{P(X_n = j) - \\pi_j} = 0.$$\nIn particular $P(X_n = j) \\to \\pi_j$ for each $j \\in S$.\n\n## Remarks\n\nThe classical proof is by [[coupling|HTR7J0W66V]].\nRun a second independent copy $(Y_n)$ started from $\\pi$; aperiodicity makes the pair $(X_n, Y_n)$ an irreducible chain with a stationary distribution.\nHence the pair is [[recurrent|PBZTBWZ4GS]] and hits the diagonal at a finite time $\\tau$, and $\\sum_j \\abs{P(X_n = j) - \\pi_j} \\le 2 P(\\tau \u003e n) \\to 0$.\nExample: on $S = \\set{1, 2}$ let $p_{12} = a$ and $p_{21} = b$ with $0 \u003c a, b \\le 1$ and $a + b \u003c 2$.\nThen $\\pi = (b, a)/(a + b)$ and $P(X_n = 1) - \\pi_1 = (1 - a - b)^n \\left(P(X_0 = 1) - \\pi_1\\right)$, which tends to $0$.\nAperiodicity is needed: for $a = b = 1$ the chain alternates, and $P(X_n = 1)$ oscillates between $0$ and $1$ when $X_0 = 1$.\nIrreducibility is needed: with two absorbing states the limit depends on the starting state.\n\n## Explanations\n\n### L1 human\n\nA chain that can reach every state from every other and does not move in a fixed cycle forgets where it started.\nAfter a long time, the chance of being in each state is close to its value under the stationary distribution, whatever the start.\nThis is why shuffling a deck long enough makes every order of the cards almost equally likely.\n\n### L2 human\n\nThe theorem justifies Markov chain Monte Carlo: running an ergodic chain produces samples whose law approaches $\\pi$.\nThe speed of convergence is measured by the [[mixing time|8TV1YW8EXC]], and for finite chains it is geometric, governed by the second largest eigenvalue modulus of $P$.\nWithout aperiodicity, the averages $\\frac1n \\sum_{k=1}^{n} P(X_k = j)$ still converge to $\\pi_j$.\nFor a finite state space, condition 3 follows from condition 1.\n",
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