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The Cantor function is a good toy example for getting a feeling of scope of the property of continuity of functions. It might challenge our intuition, showing us that the notion of continuity is broader than what we thought.

I'm looking for toy examples of this kind to gain a better intuition on stationary processes.

Could you point out to examples of stochastic processes (stationary or not) that may contribute to illustrate the scope of the stationarity dependence-structure?

For instance:

  • A stationary process whose sample-paths don't look like what we would intuitively expect from a stationary process.

  • A stationary process that is likely not to pass common stationarity tests.

  • A counter example that shows that stationarity doesn't imply a certain property.

  • Other examples in this direction.

Please make clear to which notion of stationarity you are referring, examples for any of the notions are appreciated.

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Two examples. 1.

\begin{align*} X_t &= A \cos( \omega t) + B \sin(\omega t)\\ &= 2 \cos(\omega) X_{t-1} - X_{t-2} \end{align*} where $A$ and $B$ are uncorrelated mean $0$ random variables, and $\omega \in (0, \pi)$.

This process is totally deterministic, and its autocovariance function doesn't decay (it is sinusoidal).

Example 2: the ARCH(p) model \begin{align*} Z_t &= \sqrt{h_t}e_t, \hspace{10mm} \{e_t\} \sim IID(0,1) \\ h_t &= \alpha_0 + \sum_{i=1}^p \alpha_i Z^2_{t-i}. \end{align*} $\alpha_0 > 0$, $a_i \ge 0$, $p \in \mathbb{N}$. Weakly stationary, but the conditional forecast variance changes.

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