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  "body": "\n## Definition\n\nLet $S$ be a finite set and let $P$ be the [[transition matrix|0F462493EA]] of an irreducible and aperiodic [[Markov chain|033TJBZDG0]] on $S$.\nLet $\\pi$ be its [[stationary distribution|0KRH5157WP]].\nThe *total variation distance* between probability distributions $\\mu$ and $\\nu$ on $S$ is\n$$\\norm{\\mu - \\nu}_{TV} = \\max_{A \\subseteq S} \\abs{\\mu(A) - \\nu(A)} = \\frac12 \\sum_{x \\in S} \\abs{\\mu(x) - \\nu(x)}.$$\nFor an integer $t \\ge 0$ let $d(t) = \\max_{x \\in S} \\norm{P^t(x, \\cdot) - \\pi}_{TV}$, where $P^t(x, \\cdot)$ is row $x$ of $P^t$.\nFor $\\varepsilon \\in (0, 1)$, the *mixing time* is\n$$t_{\\mathrm{mix}}(\\varepsilon) = \\min\\set{t \\ge 0 : d(t) \\le \\varepsilon},$$\nand $t_{\\mathrm{mix}} = t_{\\mathrm{mix}}(1/4)$.\n\n## Remarks\n\nThe minimum exists because $d(t) \\to 0$ by the [[convergence theorem|B1E1ES1BYD]] for ergodic chains.\nExample: on $S = \\set{1, 2}$ let every entry of $P$ be $1/2$.\nThen $\\pi = (1/2, 1/2)$, $d(0) = 1/2$ and $P^t(x, \\cdot) = \\pi$ for $t \\ge 1$, so $t_{\\mathrm{mix}} = 1$.\nNon-example: for $P = \\begin{pmatrix} 0 \u0026 1 \\\\ 1 \u0026 0 \\end{pmatrix}$ the chain is periodic, $d(t) = 1/2$ for all $t$, and $t_{\\mathrm{mix}}(\\varepsilon)$ does not exist for $\\varepsilon \u003c 1/2$.\nThe function $d$ is nonincreasing, and $t_{\\mathrm{mix}}(\\varepsilon) \\le \\ceil{\\log_2 (1/\\varepsilon)} \\, t_{\\mathrm{mix}}$.\n",
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