= Autoregressive process of order one
{title2=$\operatorname{AR}(1)$}
= AR(1)
{c}
{synonym}
An <autoregressive process of order one> obeys
$$
X_t-\mu=\phi(X_{t-1}-\mu)+\varepsilon_t,
$$
where $\phi$ is real and $\varepsilon_t$ is centered <white noise> of <variance> $\sigma^2>0$. The zero-<mean> version is $X_t=\phi X_{t-1}+\varepsilon_t$; in the affine form $X_t=c+\phi X_{t-1}+\varepsilon_t$, the stationary <mean> satisfies $c=(1-\phi)\mu$.
For $|\phi|<1$, its unique <weakly stationary process> solution on all integer times is the <causal time-series representation>
$$
X_t=\mu+\sum_{j=0}^{\infty}\phi^j\varepsilon_{t-j}.
$$
The sum converges in the sense of <mean-square convergence> because $\sum_{j\ge0}\phi^{2j}=1/(1-\phi^2)<\infty$. It is an <infinite moving-average representation> using only present and past driving noise. <Weak white noise> suffices for this second-order conclusion; <strong white noise> also gives <strict stationarity>. The <autocovariance of an AR(1) process> is $\sigma^2\phi^{|h|}/(1-\phi^2)$, so its <variance> is $\sigma^2/(1-\phi^2)$ and its <ACF> is $\phi^{|h|}$. See https://www.stat.berkeley.edu/~bartlett/courses/153-fall2010/lectures/5.pdf[Bartlett, lecture 5: AR(1) representation and causality].
The appeal of an <AR(1)> is that one coefficient expresses how a deviation persists. For $0<\phi<1$, disturbances propagate with weights $1,\phi,\phi^2,\ldots$. For $-1<\phi<0$, these weights alternate sign. With $\mu=10$, $\phi=0.7$, and $X_t=12$, the <recursive forecasts of an autoregressive process> give $11.4$ and $10.98$ one and two steps ahead. If $\sigma^2=1$, the stationary <variance> is $100/51\simeq1.9608$, while the one-step forecast error <variance> is only $1$. Under <weak white noise> these are best linear forecasts; independent driving noise also makes them <conditional expectations> given the observed past.
The condition $|\phi|<1$ describes a stationary causal solution, not every finite recursion with those coefficients. Starting from a prescribed value gives the transient in <finite initialization of an AR(1) process>. Nor is it a condition for all possible two-sided stationary solutions: when $|\phi|>1$, <noncausal stationary autoregression> uses future driving noise. See <stationary versus causal solution of a two-sided AR(1) equation> for the boundary cases.
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