what is the difference between stationary processes and processes with stationary increments?
I know that independent increments is a solid property to increments of a process, and not the process itself.
But Stationary has a different story I guess! Since we have stationary processes and process with stationary increments. It is two property that can be defined to same process.
My question is on the relation between these two properties. Does a stationary process have stationary increments or vice versa! Or these two can't be related this way or another?
I think having stationary increments is broader property and cant be narrowed down to stationary processes. I guess there are processes that are not stationary processes but have stationary increments.
Is there a way to figure out a relationship between these two?
Best Answer
Stationary processes have stationary increments. To see roughy why this is true suppose $(X_t)_{t \geq 0}$ is stationary and consider $X_{t+s}-X_t$. Since $(X_t,X_{t+s})$ has the same distribution as $(X_0,X_s)$ it follows that the increments $X_{t+s}-X_t$ and $X_s-X_0$ have the same distribution. This can be generalized to more than two increments.
The standard Brownian Motion BM $(B_t)$ has stationary increments: the distrbution of $B_s-B_t$ depends only on the difference $s-t$. But it is not stationary since the distribution of $B_t$ is not independent of $t$ (Its variance is $t$).