Weak and Strong Stationarity#

Stage 2 · 📐 Stationarity · Lesson 04 of 18 · beginner

◀ Previous · A Gentle Introduction to Stationarity · Next · Linear Processes ▶ · ↑ Section

Important

✨ AI-generated content. This page was written with the assistance of an AI language model and is provided as a learning aid. Despite careful review, it may still contain mistakes, omissions, or out-of-date information. Whether you are new to the topic, a team lead, or a senior practitioner, treat it as a starting point rather than an authoritative reference: read it critically and independently verify anything you act on (code, commands, figures, and factual claims) against official documentation and primary sources before relying on it.

Two definitions#

“Stationarity” comes in two strengths. Strong (strict) stationarity constrains the entire distribution; weak (second-order, or covariance) stationarity constrains only the first two moments. Most classical time-series theory — including ARMA — needs only the weaker one.

Strong stationarity#

A series is strictly stationary if shifting time leaves its whole joint distribution unchanged: for any lag \(h\) and any set of times, the block \((x_{t_1}, \dots, x_{t_k})\) has the same joint distribution as \((x_{t_1+h}, \dots, x_{t_k+h})\),

\[(x_{t_1}, \dots, x_{t_k}) \;\stackrel{d}{=}\; (x_{t_1+h}, \dots, x_{t_k+h}).\]

Nothing about the probabilistic structure depends on when you look — a strong condition that is hard to verify from a single finite sample.

Weak stationarity#

A series is weakly stationary if its mean is constant, its variance is finite and constant, and its autocovariance depends only on the lag — not on absolute time:

\[\mathbb{E}[x_t] = \mu, \qquad \gamma(s, t) = \operatorname{Cov}(x_s, x_t) = \gamma(|s - t|).\]

This is all that ARMA modelling requires, and — unlike strict stationarity — it is checkable from data through the sample mean and the sample autocorrelation.

How they relate#

The two coincide under mild conditions. Strong stationarity plus finite second moments implies weak stationarity. The converse fails in general, except for Gaussian processes: a Gaussian process is fully described by its mean and covariance, so weak + Gaussian implies strong. White noise — zero mean, constant variance, zero autocorrelation — is the canonical weakly stationary building block.

Hint