 Finite initialization of an AR(1) process

For $t\ge 1$, 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 $|\varphi |<1$,For $h\ge 0$, its covariance is $\mathrm{Cov}(X_{t+h},X_t)={\varphi }^h\mathrm{Var}(X_t)$, which depends on $t$. Even setting $x_0=\mu$ leaves a changing variance. At $\varphi =0.7$, ${\sigma }^2=1$, 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 ${\sigma }^2/(1-{\varphi }^2)$, and zero covariance with every future driving noise value. The full stationary construction sets $X_0-\mu ={\sum }_{j\ge 0}{\varphi }^j{\epsilon }_{-j}$. For discrete Gaussian white noise, equivalently draw $X_0$ from $N(\mu ,{\sigma }^2/(1-{\varphi }^2))$ 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.