brownian translation

Question

I have a bit of struggle with understanding what it means to have the same law as a brownian. For instance, how can i prove that $$\sup_{t\in[k-1,k]} |W_t - W_{k-1}| \stackrel{\mathcal{L}}{=} \sup_{t\in[0,1[} |W_t| $$ where $k \in \mathbb{Z}.$
My attempt at a proof is the following:
since $W_t - W_{k-1} \sim \mathcal{N}(0,t-(k-1))$, and $W_{t-(k-1)} \sim \mathcal{N}(0,t-(k-1))$ : $$W_t - W_{k-1} \stackrel{\mathcal{L}}{=} W_{t-(k-1)}$$ hence the equality above.

Answer 1

No! You need much more than just $W_t-W_{k-1}\sim N(0,t-k+1)$.

You want to show that $W_{t+k}-W_k$ is another Wiener process, i.e., you want to show

• it is almost surely $=0$ at $t=0$,
• independent Gaussian increments, and
• continuous paths with probability 1

These follows from $W_t$ being a Wiener process (why?).