Prove $V_t=B_{t+T}-B_T$ is Weiner Process

[Jason4]

1) For each $t$, find $P(B_t\neq 1)$ .

2) For any $T>0$, prove $V_t=B_{t+T}-B_T$ is a Weiner process.

...

For 2) should I be looking at something like this:

Let: $\Delta (t+T)=\frac{t}{N}$ for large $N$

$\Rightarrow B_{t+T}-B_T=(B_{T+\frac{t}{N}}-B_T)+(B_{T+\frac{2t}{N}}-B_{T+\frac{t}{N}})+\ldots+(B_{t+T}-B_{t+T-\frac{t}{N}})$

This is a sum of $N$ iid r.v.s with mean $0$ and variance $(\Delta (t+T))\sigma^2$. By the CLT we get:

$B_{t+T}-B_T\sim N(0,N\, \frac{t}{N}\,\sigma^2)=N(0,t\sigma^2)$

Now take the limit as $N \rightarrow 1$.

 

[Jhero]

We get:$\lim_{N \rightarrow \infty} B_{t+T}-B_T = N(0,t\sigma^2) \Rightarrow$ $V_t = B_{t+T}-B_T$ is a Weiner process.