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  "body": "\n## Definition\n\nLet $S$ be a nonempty [[countable set|0ZG61TFBES]].\nLet $(X_n)_{n \\ge 0}$ be a [[stochastic process|0QH5ENKZM1]] with values in $S$. It is a Markov chain with state space $S$ if\n$$P(X_{n+1} = j \\mid X_0 = i_0, \\dots, X_{n-1} = i_{n-1}, X_n = i) = P(X_{n+1} = j \\mid X_n = i)$$\nfor each $n \\ge 0$ and all states $i_0, \\dots, i_{n-1}, i, j \\in S$ with $P(X_0 = i_0, \\dots, X_n = i) \u003e 0$.\nThe chain is time-homogeneous if $p_{ij} = P(X_{n+1} = j \\mid X_n = i)$ does not depend on $n$.\n\n## Remarks\n\nThe condition is the [[Markov property|0HB8SXHPFA]] in discrete time.\nFor a time-homogeneous chain, the numbers $p_{ij}$ form the [[transition matrix|0F1KBVY161]] $P = (p_{ij})_{i, j \\in S}$.\nThe initial distribution $\\lambda_i = P(X_0 = i)$ and $P$ determine the law of the chain: $P(X_0 = i_0, \\dots, X_n = i_n) = \\lambda_{i_0} p_{i_0 i_1} \\cdots p_{i_{n-1} i_n}$.\nMarkov chains also exist in continuous time and on general state spaces.\n\n## Explanations\n\n### L1 human\n\nA Markov chain has no memory beyond the present state.\nConsider the weather each day: if tomorrow depends only on today, the sequence of weather states is a Markov chain.\n",
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  "name": "Markov chain",
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