Let be $$Z(t)= \int_t^{t+1}B(s)\,ds-B(t)$$ a process where $\{B(t), t \ge0\}$ be a standard Brownian Motion. I have to show that $Z(t)$ has stationary increments.
$\textbf{My very poor ideas: }$
I have to prove that $Z(t+h)-Z(t) \sim Z(h)-Z(0)$ for all $h \ge 0$.
I see that $Z(0) \sim \mathcal{N}(0,\frac13)$ but I don't know if this let me far away because If i extend $Z(t+h)-Z(t)$, it looks like very complicate.
So maybe can I prove that $\int_t^{t+1}B(s)\,ds$ is indipendent from $B(t)$? Is it helpful?
Thank you for any hint. Sorry for my bad english.