# Is random walk stationary or non-stationary?

I'm confused with the concept "stationarity".
Reference: https://otexts.com/fpp3/stationarity.html

I believe random walk is non-stationary and change (difference) in random walk is stationary. The characteristics of random walk are these below:

• long periods of apparent trends up or down
• sudden and unpredictable changes in direction.

On the other hand, this book mentions time series with aperiodic cyclic behavior are considered stationary. In my opinion, this "cyclic behavior" is very similar to the behavior of random walk.

Some cases can be confusing — a time series with cyclic behaviour (but with no trend or seasonality) is stationary. This is because the cycles are not of a fixed length, so before we observe the series we cannot be sure where the peaks and troughs of the cycles will be.

How should I interpret these passages?

• A random walk has no stationary distribution, hence cannot be stationary. Dec 18, 2022 at 11:21

A random walk $$\langle X_t\rangle$$ is defined as

\begin{align}X_t &:= X_{t-1} + Z_t\\ &= \sum_{j=1}^t Z_j,\tag 1\label 1\end{align}

where $$\langle Z_t\rangle$$ is a discrete time, pure random process with mean being $$\mu,$$ and variance $$\sigma^2.$$ So, from $$\eqref 1, ~\mathbb E[X_t] = \mu t; ~~\operatorname{Var}[X_t] = \sigma^2 t,$$ which imply the mean and variance vary over $$t.$$ This certainly means random walk process $$\langle X_t\rangle$$ is non-stationary.

Now $$\nabla X_t = Z_t,$$ which is definitely stationary as it is a purely random process.

Cyclical variations are intuitively speaking manifestations of oscillations not having a fixed period.

Stationarity means there is no systematic change in mean and variance: properties of a section should resemble that of other section.

So, what the authors in the given quote clearly meant is that presence of cyclical behavior needn't imply non-stationarity. Because they, by definition, aren't of fixed length.

Random walks on other hand exhibit changes in variance over time, with "sudden and unpredictable changes in direction."

## Reference:

$$\rm [I]$$ The Analysis of Time Series, Chris Chatfield, CRC Press, $$2003.$$

• pure random process does not sound like an unambiguous technical term. A martingale difference sequence would be closer to one. Moreover, a more widespread definition of a random walk would require $\mu=0$. $\mu\neq 0$ produces a random walk with a drift. Dec 18, 2022 at 14:14
• Well, purely random process refers to a sequence of $\rm i.i.d.$ random variables with mean $\mu$ and variance $\sigma^2.$ Dec 18, 2022 at 14:21
• Saw your updated comment. Yes. I followed Chatfield's treatment of the same. Dec 18, 2022 at 14:24