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  "body": "\n## Definition\n\nLet $(X_n)_{n \\ge 0}$ be a time-homogeneous [[Markov chain|033TJBZDG0]] on a countable state space $S$.\nFor a state $i \\in S$, write $P_i$ and $E_i$ for probability and expectation when $X_0 = i$.\nLet $T_i = \\inf\\set{n \\ge 1 : X_n = i}$ be the first return time to $i$, with $\\inf \\emptyset = \\infty$.\n\n1. The state $i$ is *recurrent* if $P_i(T_i \u003c \\infty) = 1$.\n2. The state $i$ is *transient* if $P_i(T_i \u003c \\infty) \u003c 1$.\n3. A recurrent state $i$ is *positive recurrent* if $E_i[T_i] \u003c \\infty$, and *null recurrent* if $E_i[T_i] = \\infty$.\n\n## Remarks\n\nLet $p^{(n)}_{ii}$ be the diagonal entries of the $n$-th power of the [[transition matrix|0F462493EA]].\nThen $i$ is recurrent if and only if $\\sum_{n \\ge 1} p^{(n)}_{ii} = \\infty$, and this sum is the expected number of returns to $i$.\nExample: for the simple [[random walk|04VZE3FZXF]] on $\\Z$, $p^{(2n)}_{00} = \\binom{2n}{n} 4^{-n}$, which behaves like $1/\\sqrt{\\pi n}$.\nThe sum diverges, so $0$ is recurrent; it is null recurrent, since this walk has no [[stationary distribution|0KRH5157WP]].\nNon-example: for the walk on $\\Z$ that steps $+1$ with probability $p \\ne 1/2$ and $-1$ with probability $q = 1 - p$, one has $p^{(2n)}_{00} = \\binom{2n}{n} (pq)^n \\le (4pq)^n$.\nSince $4pq \u003c 1$, the sum converges and $0$ is transient, so clause 1 fails.\nIn an irreducible chain, either all states are transient, or all are null recurrent, or all are positive recurrent.\nAn irreducible chain has a stationary distribution if and only if its states are positive recurrent.\n",
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