Definition of Stationary Increment

Question

The definition of a process with stationary increments is that $$\forall h\in \mathbb{R}_+,\forall t<s,f(s)-f(t)=^d f(s+h)-f(t+h).$$ However, the definition of a stationary process is that $$\forall h\in \mathbb{R}_+,\forall t_1<\cdots<t_n,\big(f(t_1),\cdots,f(t_n)\big)=^d\big(f(t_1+h),\cdots,f(t_n+h)\big)$$ Hence, I wonder if the stationarity of the increments is just a weak version for the stationary process or though they have different dimensions, they are equivalent due to the special properties of increments.

Answer 1

This also confused me. But I think the formulation is due to that one can think of this as a process in $h$. Then one considers the finite dimensional distribution of that, and should be able to translate that by any constant which kind of get absorbed into $t$, i.e

$(f(h_{1}),\ldots,f(h_{n}))=^{d}(f(h_{1}+h),\ldots,f(h_{n}+h))=^{d}(f(h_{1}+t),\ldots,f(h_{n}+t))$

by just thinking of the translation as in $t$ and then starting from any $s$ juts translate the argument into $(f(s+h_{1}),\ldots,f(s+h_{n}))=(f(s+h_{1}+t),\ldots,f(s+h_{n}+t))$.

Which I think is what the defintion say.