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Formal Proof of the General Stationarity Condition for an AR(p) Process (Roots of Characteristic Polynomial)

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The Formal Theorem

An AR(p) process defined by Yt=c+∑i=1pϕiYt−i+ϵt Y_t = c + \sum_{i=1}^p \phi_i Y_{t-i} + \epsilon_t is stationary if and only if all the roots of its characteristic polynomial, Φ(z)=1−ϕ1z−ϕ2z2−⋯−ϕpzp=0 \Phi(z) = 1 - \phi_1 z - \phi_2 z^2 - \dots - \phi_p z^p = 0 , lie strictly outside the unit circle. That is, for every root zk z_k , ∣zk∣>1 |z_k| > 1 .

Analytical Intuition.

Visualize the AR(p) process as a system with a memory. The stationarity condition is akin to ensuring this memory doesn't accumulate uncontrollably, leading to explosive behavior. The characteristic polynomial's roots are the system's 'natural frequencies' or 'decay rates'. If any root has a magnitude less than or equal to one, its corresponding component in the system's response will either persist indefinitely (magnitude = 1) or grow over time (magnitude < 1), destabilizing the process. We demand all roots be 'faster' than this, decaying away so that the system always returns to its mean, like a well-damped pendulum settling after a push.
CAUTION

Institutional Warning.

Students sometimes confuse the roots of the characteristic polynomial of the AR polynomial with those of the MA polynomial, which have opposite conditions for stationarity/invertibility.

Academic Inquiries.

01

What does it mean for roots to lie 'strictly outside the unit circle'?

It means the modulus (or absolute value) of each complex root zk z_k must be strictly greater than 1. For real roots, this means the root is either greater than 1 or less than -1.

02

Why is the characteristic polynomial relevant to stationarity?

The characteristic polynomial arises from the homogeneous part of the AR(p) recurrence relation. Its roots determine the nature of the 'homogeneous solution,' which describes the unforced behavior of the process. If this unforced behavior grows unboundedly, the process is non-stationary.

03

Can we prove this without relying on complex analysis or spectral theory?

Yes, though spectral theory provides the most elegant proof. Recursive substitution and induction can demonstrate how roots outside the unit circle lead to decaying terms in the expression for Yt Y_t , while roots inside or on the unit circle lead to persistent or growing terms.

04

What happens if a root is exactly on the unit circle?

If a root has a magnitude of exactly 1, the corresponding component of the process will oscillate and neither decay nor grow unboundedly. This is still considered non-stationary, as the variance will not be finite.

Standardized References.

  • Definitive Institutional SourceBrockwell, Peter J., and Richard A. Davis. Introduction to Time Series and Forecasting.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). Formal Proof of the General Stationarity Condition for an AR(p) Process (Roots of Characteristic Polynomial): Visual Proof & Intuition. Retrieved from https://nicefa.org/library/time-series-analysis/formal-proof-of-the-general-stationarity-condition-for-an-ar-p--process--roots-of-characteristic-polynomial-

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