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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.

Take the latent autoregression as the scalar state $S_t$. A state-space model isThe transition and observation matrices are both scalar, $F=\varphi$, $H=1$. State-noise variance is $Q={\sigma }_z^2$, observation-noise variance is $R={\sigma }_w^2$, and the cross-noise covariance is zero at every pair of times.

A complete stationary initialization isIt is orthogonal to future state noise and to all observation noise. This is the stationary initialization of a scalar linear state-space model. This specifies the initial state in terms of the actual given two-sided white noise sequence, as well as its second-order law; simply starting from zero would give transient rather than stationary observations.

If a Gaussian state-space specification is intended, the complete specialization is $S_0\sim N(0,Q/(1-{\varphi }^2))$, independent of the future iid Gaussian state and observation noises, themselves independent with variances $Q,R$. Under the printed assumptions alone, Gaussian distributions and independence cannot be deduced from white noise orthogonality; the equations and stationary-series initialization above give the exact second-order representation without adding them.