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Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1)

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The Formal Theorem

Let {Yt} \{Y_t\} be an Autoregressive process of order 1 (AR(1)) defined by the linear recurrence relation:
Yt=c+ϕYt−1+ϵt Y_t = c + \phi Y_{t-1} + \epsilon_t
where c c is a real constant, ϕ \phi is the autoregressive parameter, and {ϵt} \{\epsilon_t\} is a white noise process satisfying the following conditions: 1. E[ϵt]=0 E[\epsilon_t] = 0 for all t t . 2. Var[ϵt]=σϵ2<∞ Var[\epsilon_t] = \sigma^2_{\epsilon} < \infty for all t t . 3. Cov[ϵt,ϵs]=0 Cov[\epsilon_t, \epsilon_s] = 0 for all t≠s t \neq s . 4. ϵt \epsilon_t is uncorrelated with past values of the process, i.e., Cov[ϵt,Ys]=0 Cov[\epsilon_t, Y_{s}] = 0 for all s<t s < t . **Theorem:** An AR(1) process {Yt} \{Y_t\} is wide-sense stationary if and only if ∣ϕ∣<1 |\phi| < 1 . Under this stationarity condition, the process {Yt} \{Y_t\} can be expressed as a causal, absolutely convergent infinite moving average (MA(∞\infty)) process, and its key statistical moments are: 1. **Constant Mean:** E[Yt]=c1−ϕ E[Y_t] = \frac{c}{1 - \phi} 2. **Constant Finite Variance:** Var[Yt]=σϵ21−ϕ2 Var[Y_t] = \frac{\sigma^2_{\epsilon}}{1 - \phi^2} 3. **Autocovariance Function (depends only on lag k k ):** Cov[Yt,Yt−k]=σϵ21−ϕ2ϕ∣k∣ Cov[Y_t, Y_{t-k}] = \frac{\sigma^2_{\epsilon}}{1 - \phi^2} \phi^{|k|} for any integer k k

Analytical Intuition.

Imagine a complex ecosystem, where Yt Y_t represents the population of a key species at time t t . The parameter ϕ \phi acts as an inherent feedback mechanism: how much the previous generation's size Yt−1 Y_{t-1} influences the current one. If ∣ϕ∣≥1 |\phi| \ge 1 , this feedback is explosive or self-sustaining. Each generation's influence on the next is either amplified or remains undiminished, leading to wild, uncontrolled population booms or busts, or an indefinite drift. The ecosystem never finds balance; its average population or variability constantly shifts, making it non-stationary.
However, if ∣ϕ∣<1 |\phi| < 1 , the feedback mechanism is a dampener. The influence of previous generations Yt−1 Y_{t-1} gradually fades with each passing season. The ϵt \epsilon_t term introduces random environmental shocks (e.g., a sudden resource boom), but its impact is also progressively diluted. This crucial dampening ensures that the ecosystem's population eventually settles into a stable, predictable rhythm. Its average size and fluctuations remain constant over time, achieving stationarity—a state where the system's fundamental statistical character endures.
CAUTION

Institutional Warning.

Students often struggle to fully grasp why ∣ϕ∣<1 |\phi| < 1 is not just a sufficient condition, but a necessary one for a unique stationary solution. They might also misattribute properties of white noise (e.g., normality) as prerequisites for wide-sense stationarity, when only its first two moments and lack of autocorrelation are essential.

Academic Inquiries.

01

Why do we typically focus on wide-sense stationarity rather than strict stationarity for AR(1) processes?

Strict stationarity is a very demanding condition, requiring the joint probability distribution of any set of observations (Yt,…,Yt+k) (Y_t, \dots, Y_{t+k}) to be identical to (Yt+h,…,Yt+k+h) (Y_{t+h}, \dots, Y_{t+k+h}) for all t,k,h t, k, h . This is often difficult to prove or verify. Wide-sense (or covariance) stationarity, which only requires constant mean, constant finite variance, and autocovariance depending solely on the lag, is much more mathematically tractable. For linear processes like AR(1) with Gaussian white noise errors, wide-sense stationarity actually implies strict stationarity, making it a powerful and practically relevant concept.

02

What happens if ϕ=1 \phi = 1 or ∣ϕ∣>1 |\phi| > 1 ?

If ϕ=1 \phi = 1 , the process becomes a random walk (if c=0 c=0 ) or a random walk with drift (if c≠0 c \neq 0 ). In this scenario, the variance Var[Yt]=σϵ21−ϕ2 Var[Y_t] = \frac{\sigma^2_{\epsilon}}{1 - \phi^2} becomes infinite, as the denominator tends to zero. The process does not revert to a mean, and its values can wander indefinitely, so it is non-stationary. If ∣ϕ∣>1 |\phi| > 1 , the process is 'explosive.' Past shocks (ϵt \epsilon_t ) are amplified exponentially over time, causing the variance to grow without bound, also resulting in a non-stationary process. Such processes lack stable statistical properties.

03

How does the white noise term ϵt \epsilon_t influence stationarity?

The white noise term ϵt \epsilon_t serves as a continuous input of uncorrelated shocks or innovations into the system at each time step. While it introduces randomness, its specific properties (zero mean, constant finite variance, and zero autocorrelation) are critical. It ensures that new information enters the system without accumulating or creating its own feedback loops. When ∣ϕ∣<1 |\phi| < 1 , the dampening effect of ϕ \phi ensures that the influence of these past shocks gradually fades, allowing the process to settle into a stationary state where its statistical characteristics remain invariant over time.

Standardized References.

  • Definitive Institutional SourceBrockwell, P. J., & Davis, R. A. (2016). Introduction to Time Series and Forecasting. 4th ed. Springer.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Proof of the Stationarity Condition for an AR(1) Process (|φ| < 1): Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/proof-of-the-stationarity-condition-for-an-ar-1--process--------1-

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