A stochastic process organizes a family of random variables by an index. When the index is time, a basic question is whether the probability law changes over time. For a spatial index, the corresponding question concerns shifts in position.

Stationarity describes statistical laws that remain invariant under time shifts. It does not require a sample path to remain constant.

1. Definitions of Stationarity

1.1 An obviously nonstationary process

Consider Xt=t+ϵtX_t=t+\epsilon_t, where ϵt∼N(0,1)\epsilon_t\sim N(0,1). This process has E[Xt]=tE[X_t]=t, so the center of its distribution moves to the right over time. X10X_{10} fluctuates around 10, while X100X_{100} fluctuates around 100. Their statistical laws differ, making the process nonstationary.

1.2 Invariance under time shifts

Take a segment of a process, Xt,Xt+1,Xt+2X_t,X_{t+1},X_{t+2}, and shift the observation window forward by 100 units to Xt+100,Xt+101,Xt+102X_{t+100},X_{t+101},X_{t+102}.

If the two segments have the same statistical law, the process has time-shift invariance. Stationarity requires the laws to match, not the realized values.

1.3 Strict stationarity

A process is strictly stationary if, for every collection t1,…,tnt_1,\dots,t_n and every shift τ\tau, (Xt1,…,Xtn)(X_{t_1},\dots,X_{t_n}) and (Xt1+τ,…,Xtn+τ)(X_{t_1+\tau},\dots,X_{t_n+\tau}) have exactly the same joint distribution. No finite collection changes its joint probability structure under a common shift. This is a strong condition because it applies to joint distributions of every order.

1.4 Weak stationarity

Applications often use the weaker condition of weak stationarity, also called second-order stationarity. It normally requires:

  1. a time-independent mean, E[Xt]=μE[X_t]=\mu;
  2. covariance that depends only on separation, not absolute time:
Cov⁡(Xt,Xt+τ)=C(τ).(1)\operatorname{Cov}(X_t,X_{t+\tau})=C(\tau). \tag{1}

Any two points separated by the same lag have the same covariance. A function of two times s,ts,t can therefore be reduced to a function of one lag τ\tau.

2. Common Misconceptions and Typical Processes

2.1 Stationary does not mean motionless

A stationary process can change sharply at every time point. For example, i.i.d. white noise satisfies Xt∼N(0,1)X_t\sim N(0,1), with independent values across time. A sample path might be 0.3 → -1.7 → 2.1 → 0.4 → -0.9 → ....

The values continually change, but every time has the same distribution and the joint structure is shift invariant. The process is stationary.

Stationarity describes probability laws, not the smoothness of a sample path.

2.2 Why white noise matters

White noise usually satisfies:

  • E[ϵt]=0E[\epsilon_t]=0;
  • Var⁡(ϵt)=σ2\operatorname{Var}(\epsilon_t)=\sigma^2;
  • values at different times are uncorrelated, and many models also assume independence.

White noise has no trend or visible temporal memory. Many complex processes can be understood as transformations of white noise. A random walk, Xt=∑i=1tϵiX_t=\sum_{i=1}^t\epsilon_i, accumulates a sequence of white-noise increments.

2.3 Stationary processes and stationary increments

A random walk is usually not stationary. If each ϵt\epsilon_t has variance σ2\sigma^2, then Var⁡(Xt)=tσ2\operatorname{Var}(X_t)=t\sigma^2.

The position distribution spreads out as tt grows, so the law of XtX_t changes.

If the distribution of Xt+h−XtX_{t+h}-X_t depends only on hh and not on tt, the process has stationary increments.

  • A stationary process has a statistical law invariant under time shifts.
  • A process with stationary increments has the same distribution of changes over intervals of equal length.

Brownian motion is not itself stationary, but it has stationary increments.

3. The Value and Limits of Stationarity

3.1 Stationarity supports inference from past to future

If the data-generating mechanism continually changes, statistics estimated from past observations may quickly become invalid. Stationarity allows estimated laws from the past to describe comparable future intervals.

Classical time-series analysis therefore often begins by testing whether a series is stationary. A series with a trend, seasonality, or structural breaks may require differencing, detrending, or another transformation.

3.2 Stationarity does not imply convergence to a fixed value

A stationary process may fluctuate forever. A nonstationary process may also reveal stable laws after a suitable transformation.

A stationary distribution for a Markov chain is related to, but distinct from, a stationary process. A distribution π\pi is stationary when it remains unchanged after a Markov transition: πP=π\pi P=\pi.

4. Core Concepts

Strict stationarity preserves every joint distribution under time shifts. Weak stationarity requires a constant mean and covariance that depends only on lag.

Two distinctions are essential: a stationary sample path can vary dramatically, and stationarity of a process differs from stationarity of its increments. Stationarity describes stable long-run statistical structure; the laws of large numbers and central limit theorems explain why stable patterns emerge when many random variables accumulate.