Autoregressive process of order one

{title2=$X_t-\mu=\phi^t(X_0-\mu)+\sum_{j=0}^{t-1}\phi^j\varepsilon_{t-j}$}

For $t\ge1$, iterating an <AR(1)> from a fixed value $X_0=x_0$ gives the displayed finite sum. With centered <white noise> of <variance> $\sigma^2$ and $|\phi|<1$,
$$
\mathbb E[X_t]=\mu+\phi^t(x_0-\mu),\qquad
\operatorname{Var}(X_t)=\frac{\sigma^2(1-\phi^{2t})}{1-\phi^2}.
$$
For $h\ge0$, its <covariance> is $\operatorname{Cov}(X_{t+h},X_t)=\phi^h\operatorname{Var}(X_t)$, which depends on $t$. Even setting $x_0=\mu$ leaves a changing <variance>. At $\phi=0.7$, $\sigma^2=1$, and $x_0=\mu$, the <variances> at times zero through three are $0,1,1.49,1.7301$, approaching $100/51$; this finite initialized record is not weakly stationary.

To initialize in <weak stationarity>, give $X_0$ <mean> $\mu$, <variance> $\sigma^2/(1-\phi^2)$, and zero <covariance> with every future driving noise value. The full stationary construction sets $X_0-\mu=\sum_{j\ge0}\phi^j\varepsilon_{-j}$. For <discrete Gaussian white noise>, equivalently draw $X_0$ from $N(\mu,\sigma^2/(1-\phi^2))$ independently of future noise. This gives exact <strict stationarity> from the first plotted point; it is the initialization used in the figure in <time series>. Matching these first two moments alone does not guarantee <strict stationarity> for non-Gaussian inputs. See also <stationary initialization of a scalar linear state-space model>.


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 Incoming links: Finite initialization of an AR(1) process