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Series 1 wanders over a changing level rather than fluctuating around a stable local mean. Its sample autocorrelation function is strongly positive and decreases very slowly. This is the usual diagnostic evidence for an ordinary unit root: an autoregressive polynomial containing $1-z$, with a zero at $z=1$, and a stationary model after first differencing. The plots support an integrated model, rather than specifying the number of its remaining stationary autoregressive or moving-average terms.

Series 2 has a pronounced oscillation with period about six observations. Its sample autocorrelation alternates between large positive and negative values with little damping: approximately positive at multiples of six and negative halfway between. Together with the changing amplitude, this suggests a conjugate pair of unit-circle zeros nearThe associated real autoregressive factor is $1-2\cos (\pi /3)z+z^2=1-z+z^2$. A targeted filter $1-B+B^2$ removes this pair; the broader seasonal difference operator $1-B^6$ also contains it but introduces additional differencing factors. This is the oscillatory unit-root diagnosis from an undamped sample autocorrelation.

Thus Series 1 suggests a zero at 1; Series 2 suggests a conjugate pair on the unit circle at a seasonal frequency. These are model diagnoses, not deductions of exact roots from a finite sample. A stationary model very close to a unit root can look similar, and an undamped periodic covariance can also arise from a stationary random sinusoid. The figure does not identify exact orders or prove nonstationarity by itself.

Interpret stationarity in the usual second-order time-series sense and assume nondegenerate noise, ${\sigma }_z^2>0$. For a two-sided autoregressive equation, the missing existence condition isIt is important to separate this from causality. If $|\varphi |<1$, the unique stationary solution is $X_t={\sum }_{j\ge 0}{\varphi }^j{Z}_{t-j}$. If $|\varphi |>1$, there is still a stationary solution, but it is anticausal:Both expansions converge in L2 because their coefficients are square summable. Substitution verifies the equation. Their means are zero and their covariance functions depend only on lag. Uniqueness follows by iterating the equation backward in the first case and forward in the second: the remainders ${\varphi }^n{X}_{t-n}$ or ${\varphi }^{-n}{X}_{t+n}$ tend to zero in L2 for any stationary finite-variance solution. This is the stationary versus causal solution of a two-sided AR(1) equation.

For $\varphi =\pm 1$, iteration givesThe variance of the right side is $n{\sigma }_z^2$. The variance of the left side is at most $4\mathrm{Var}(X_t)$ by stationarity and Cauchy-Schwarz inequality. These are incompatible as $n\to \infty$. Thus no weakly stationary finite-variance solution exists at those unit roots.

If the intended claim includes a causal innovation representation, its condition is instead $|\varphi |<1$, as in the next part. The stated white noise equation alone does not say that $Z_t$ is orthogonal to the past of $X$. If zero innovation variance is allowed, the unit-root exclusion has degenerate exceptions, such as random constant solutions when $\varphi =1$; the nondegenerate convention is necessary for the asserted nonexistence.