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  "body": "\n## Definition\n\nA *white noise* with variance $\\sigma^2 \u003e 0$ is a sequence $(\\varepsilon_t)_{t \\in \\Z}$ of real random variables with $E[\\varepsilon_t] = 0$, $E[\\varepsilon_t^2] = \\sigma^2$ and $E[\\varepsilon_s \\varepsilon_t] = 0$ for $s \\ne t$.\nLet $p \\ge 1$, let $c, \\phi_1, \\dots, \\phi_p \\in \\R$ with $\\phi_p \\ne 0$, and let $(\\varepsilon_t)$ be a white noise.\nAn *autoregressive model of order $p$*, or AR($p$), is a weakly [[stationary process|DT0QXJE705]] $(X_t)_{t \\in \\Z}$ that satisfies\n$$X_t = c + \\phi_1 X_{t-1} + \\cdots + \\phi_p X_{t-p} + \\varepsilon_t \\qquad \\text{for all } t \\in \\Z.$$\nIts *autoregressive polynomial* is $\\phi(z) = 1 - \\phi_1 z - \\cdots - \\phi_p z^p$.\n\n## Remarks\n\nIf $\\phi(z) \\ne 0$ for all complex $z$ with $\\abs{z} = 1$, the equation has exactly one weakly stationary solution.\nIf moreover $\\phi$ has no [[root|019MA1VTR2]] in the closed unit disk $\\abs{z} \\le 1$, this solution is causal: $X_t = \\mu + \\sum_{j \\ge 0} \\psi_j \\varepsilon_{t-j}$ with $\\sum_j \\abs{\\psi_j} \u003c \\infty$ and $\\mu = c / (1 - \\phi_1 - \\cdots - \\phi_p)$.\nMany texts include causality in the definition.\n\nExample: for $p = 1$, $c = 0$ and $\\abs{\\phi_1} \u003c 1$, the process $X_t = \\sum_{j \\ge 0} \\phi_1^j \\varepsilon_{t-j}$ converges in mean square and satisfies $X_t = \\phi_1 X_{t-1} + \\varepsilon_t$.\nIts variance is $\\sigma^2 / (1 - \\phi_1^2)$ and its autocorrelation at lag $h$ is $\\phi_1^{\\abs{h}}$.\n\nNon-example: for $\\phi_1 = 1$ and $c = 0$, the equation $X_t = X_{t-1} + \\varepsilon_t$ defines a [[random walk|04VZE3FZXF]], and it has no weakly stationary solution.\nIndeed $X_t - X_{t-n} = \\varepsilon_{t-n+1} + \\cdots + \\varepsilon_t$ has variance $n \\sigma^2$, which is unbounded in $n$, while for a stationary solution it is at most $4 \\operatorname{Var}(X_0)$.\nHere $\\phi(z) = 1 - z$ has the root $z = 1$ on the unit circle.\n",
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