Can a stationary process have a nonzero mean?

Yes. The worked example has mean 20.

Can observations be negatively correlated?

Yes. Stationarity permits lag covariance of either sign when the overall covariance structure is valid.

Does a stationary process always have a finite variance?

Strict stationarity alone does not require it; weak stationarity does.

AI Glossary / Time series and signal processing / Advanced

Stationary process

Definition

A stationary process is a stochastic process whose statistical behavior is invariant under shifts in time, according to a specified definition of stationarity. Strict stationarity concerns all finite joint distributions. Weak stationarity concerns a constant finite mean and a covariance structure depending only on lag, with finite variance. A stationary process can still produce changing, correlated observations.

What it means

The process is the probabilistic mechanism; a recorded time series is one realization of it. Different realizations can look different even when they follow exactly the same stationary mechanism. A short run of high values therefore does not automatically prove a change in the process. Equally, a stable-looking average does not establish that variability and dependence have remained stable.

How it works

A useful construction starts with identically distributed independent innovations and applies the same finite rule at every time. Because shifting time does not change the joint law of the innovations, the output process is strictly stationary. If its second moments are finite, it is also weakly stationary.
For weak stationarity, compute or estimate the mean and lag covariances. Dependence is allowed provided its structure does not change with calendar time. Stationarity alone does not ensure that averages from one long realization recover ensemble averages; that requires additional conditions associated with ergodicity or suitable dependence restrictions. These distinctions matter when turning a mathematical process into a statistical model for observed data.

Three practical examples

An analyst models short-memory fluctuations around a stable operating level. Neighboring values can share a disturbance without requiring the level or variance to drift over time.
A simulation team generates many independent stationary sequences to test a monitoring procedure. Repeating the simulation helps distinguish ordinary variability between realizations from a deliberate process change.
A forecasting team checks whether a fitted residual process has stable properties over the intended evaluation period. Apparent stability in the residuals is investigated separately from trends in the original measurements. These are modeling examples, not claims that a particular business process is stationary.

Think of it this way

Imagine a machine that follows the same recipe each day but uses fresh random ingredients. Its output changes, and yesterday's ingredient can influence today if the recipe says so. Stationarity means the recipe's probabilistic behavior is unchanged when the calendar shifts, not that every day's output is identical.

Worked example

Construct independent innovations ε_t, each equal to -2 or +2 with probability one-half, for all integer times. Define X_t = 20 + ε_t + 0.5 ε_(t-1). The rule is the same at every time, so this finite moving-average construction is strictly stationary.
The four equally likely innovation pairs give output values 17, 19, 21 and 23. Their mean is 20. Squared deviations are nine, one, one and nine, so variance is (9 + 1 + 1 + 9)/4 = five. These are population calculations from the stated probabilities, not estimates from four observed days.
Adjacent outputs share ε_t: it enters X_t with coefficient one and X_(t+1) with coefficient 0.5. Since the innovation variance is four, lag-one covariance is 0.5 times four = two. Lag-one correlation is 2/5 = 0.4. Outputs separated by at least two steps use disjoint independent innovations, so they are independent in this construction and their covariance is zero.
Consider the average of three consecutive outputs. Its variance is the sum of three individual variances and twice the pairwise covariances, divided by nine. That gives [15 + 2(2 + 2 + 0)]/9 = 23/9, approximately 2.5556. Incorrectly assuming independence would give 15/9, approximately 1.6667. Stationarity does not justify ignoring dependence when assessing uncertainty in an average.
For a contrasting stationary construction, choose one random variable U equal to -2 or +2 with equal probabilities and set Y_t = 20 + U at every time. Its entire path is constant at either 18 or 22. It is strictly stationary, but the time average of one path remains 18 or 22 forever rather than converging to the ensemble mean of 20. This demonstrates why stationarity alone is insufficient to justify every time-average estimate.

Illustrative case study

Fictional Cedar Monitoring uses the moving-average construction in a training exercise. An analyst computes uncertainty for a three-period average as though the observations were independent. A colleague identifies the shared innovations and recalculates the variance as 23/9.
The team adds a note that a stationary model can still require dependence-aware uncertainty calculations. It then uses the constant-random-level example to challenge the assumption that collecting a longer record always resolves uncertainty about a population mean. Before applying either lesson operationally, it plans to examine the actual dependence and data-generating conditions. This scenario illustrates statistical reasoning rather than validated performance of a monitoring service.

Common misconceptions and mistakes

Stationary does not mean constant, independent or Gaussian. Nor does a stationary marginal distribution alone establish strict stationarity, which includes relationships among multiple times. A process and its observed realization are different objects: a finite chart cannot prove the full probabilistic property. Do not equate a moving-average process driven by innovations with simply displaying a rolling average of observed data, even though the underlying mathematical ideas are related.

Business use

State which process is assumed stationary, over what operating conditions, and in which sense. Preserve temporal order when investigating dependence and evaluating forecasts. When estimating uncertainty, account for correlations rather than automatically applying independent-sample formulas. Reassess the model after changes in operations or measurement practices. The purpose of the assumption is to support defensible reuse of historical information, not to declare that the organization or environment can never change.

Check your understanding

Why is the variance of the three-period average larger than five divided by three?

Answer: Adjacent outputs are positively correlated. The two lag-one covariance terms add uncertainty that the independent-observation calculation omits.

Related terms

Stationarity
Stochastic process
Realization
White noise
innovation
moving-average process
autocovariance
ergodicity

Learn next

Replace the coefficient 0.5 by -0.5 in the constructed process. The marginal variance remains five, but lag-one covariance becomes minus two and correlation becomes -0.4. The three-period average now has variance [15 - 8]/9 = 7/9. The unchanged marginal distribution therefore does not imply unchanged averaging uncertainty.

Marginal distribution

Sources

Peter Bartlett, UC Berkeley, Introduction to Time Series Analysis, Lecture 2, stationarity and moving-average processes: https://www.stat.berkeley.edu/~bartlett/courses/fall2007/lectures/2.pdf Richard Lockhart, Simon Fraser University, STAT 804 Lecture 2, stationarity and assumptions for estimation: https://www.sfu.ca/~lockhart/richard/804/01_3/lectures/02/web.html The discrete innovation examples, calculations and Cedar Monitoring scenario are original teaching constructions.

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