I have an interesting, though not very well formulated question. Does there exist non-trivial time series stochastic model, fulfilling that his plot "looks the same" in any time interval?
More descripting: Let $(X_t, t\in\mathbb{R})$ be stochastic process (which I am interested in). Look at his realizations on [0,1], and realizations on [0,1000]. Draw their plots without axis. Then it is not possible to distinguish if the realization is on time interval [0,1] or on [0,1000].
For example, if we have standard brownian motion, it is easy to distinguish this because there will be much smaller dependence between close variables on [0,1000] than on [0,1]. In other words, graph will look "more jumpy" on [0,1000].
I suppose the model will need to have heavy tails, but it is only my intuition. Does such model exist?