Wiener process and independent increments

Question

I am trying to prove $\bar{W_t}=cW_{tc^{-2}}$ is a Wiener process when we know $W_t$ is. I have proved most properties but I am confused how to show that is has independent increments.

What I have tried:

$\bar{W_u}=cW_{uc^{-2}}$, $\bar{W_v}=cW_{vc^{-2}}$

Then:

$\bar{W_u}-\bar{W_v}=cW_{uc^{-2}}-cW_{vc^{-2}}$

$\bar{W_u}-\bar{W_v}=c(W_{uc^{-2}}-W_{vc^{-2}})$

But I don't understand how to actually show this means it has independent increments. I know $W_t$ has it but I have a factor of $c^{-2}$ with it so surely that will effect it?

Answer 1

Let $u_1<u_2<\cdots<u_n$. We have to show that $(c(W_{u_{i+1}c^{-2}}-W_{u_ic^{-2}}))$ are independent. This is immediate since $u_1c^{-2}<u_ic^{-2}<\cdots u_nc^{-2} $ and BM has independent increments.