Is stationarity of data necesarry in order to do any statistics? If we assume we have a stochastic process $X_t$ for which we have,
$$\mathbb{E}[X_t] = \mu(t) $$
$$\operatorname{Cov}(X_t,X_s) = \gamma(s,t) $$
where the dependence of the functions on $s,t$ are non-trivial.
Is it now impossible to perform any  statistical inference, or machine learning on this process, because some of it's defining features vary over time?
Do these processes with these kind of properties occur often in reality?
 A: One generalization of stationarity is intrinsic stationarity, i.e. for a process $X_t$ we define the increments
$$
Z^{(h)}_t := X_{t+h} - X_t
$$
and require $Z^{(h)}$ to be stationary in $t$ for every $h$. You can then define the "variogram":
$$
\gamma (h) := \text{Var}(Z_t^{(h)}) = \text{Var}(Z_0^{(h)})
$$
where the last equality is due to stationarity of $Z^{(h)}$. The most prominent member of this class of stochastic process is the Brownian Motion. Also all Levy processes fit, as they have (independent) identically distributed increments. Note that we do not really need/want independence. So the entire class of Lévy processes is just a small subset of these types of processes.
If $X_t$ is stationary, then it is also intrinsically stationary with
$$
\gamma(h) = C(0) - C(h)
$$
where $C(h) = \text{Cov}(X_{t+h}, X_t)$.
But I am not really aware of a meaningful generalization beyond that. If anyone does, please tell me - that would currently be useful to my research.
The thing is: you generally have to estimate the covariance structure somehow. If you have identically distributed random variables available, then you have a shot. So either you want $X_t$ and $X_s$ to be identically distributed, or at least the increments. Otherwise you need multiple realizations of the same process which you usually do not have.
