Let $X_t$ be a stochastic process on the reals. I was under the impression the definition of stationary increments was that $X_{t+s}-X_{s}$ has the same distribution as $X_{t}-X_{0}$ for all $s,t$. I also thought stationary processes have stationary increments (is this correct)?
However, the wikipedia page for Levy process (https://en.wikipedia.org/wiki/Lévy_process) says:
'Stationary increments: For any $s<t$, $X_t-X_s$ is equal in distribution to $X_{t-s}$.
Suppose now $X_t$ is a Gaussian Process with mean $0$ and covariance function $K(t-s), K(0)=1$. By construction $X_t$ is stationary and therefore has stationary increments. However, $X_t-X_s$ has distribution $N(0,2-2K(t-s))$, whereas $X_{t-s}$ has distribution $N(0,1)$, and they are not equal in general?
Which definition is the correct one? I'm guessing $X_t-X_s$ is equal in distribution to $X_{t-s}-X_{0}$ would be the correct definition but the wikipedia page omitted $X_{0}$ because it's assumed to be almost surely 0 for Levy processes?
Stationary processes have stationary increments: Joint distribution of $X_0,X_t$ is same as that of $X_s, X_{t+s}$. This implies $f(X_0,X_t)$ has the same distribution as $f(X_s,X_{t+s})$ for any measurable function $f:\mathbb R^{2} \to\mathbb R$. Take $f(x,y)=x-y$.
You are right about the Wikipedia definition. They seem to assume that $X_0=0$. Otherwise, you have to change $X_{t-s}$ to $X_{t-s}-X_0$ in that definition.