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  "body": "\n## Statement\n\nLet $(X_n)_{n \\ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$. For $n \\ge 0$, let $p^{(n)}_{ij} = P(X_{m+n} = j \\mid X_m = i)$ be the $n$-step transition probabilities.\nThen for all $m, n \\ge 0$ and $i, j \\in S$,\n$$p^{(m+n)}_{ij} = \\sum_{k \\in S} p^{(m)}_{ik} \\, p^{(n)}_{kj}.$$\nIn matrix form, $P^{(m+n)} = P^{(m)} P^{(n)}$, so $P^{(n)} = P^n$ for the [[transition matrix|0F1KBVY161]] $P$.\n\n## Remarks\n\nProof: condition on the state $X_m = k$ at the intermediate time, with the [[law of total probability|0E5G5FMT2R]], and use the [[Markov property|0HB8SXHPFA]].\nThe equation says that to go from $i$ to $j$ in $m + n$ steps, the chain must be at some state $k$ after $m$ steps.\nFor a Markov process with transition kernels $p_{s,t}(x, dy)$, the equation reads $p_{s,u}(x, A) = \\int p_{s,t}(x, dy) \\, p_{t,u}(y, A)$ for $s \\le t \\le u$.\n\n## History\n\nThe equation is named after Sydney Chapman and Andrey Kolmogorov.\n\n## Explanations\n\n### L1 human\n\nThe chances for several steps come from multiplying the one-step transition matrix with itself.\nFor example, the chance of rain in two days, given sun today, sums over the two possible states of tomorrow.\n",
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