I am to prove that, given a Brownian Motion(Wiener Process) $\{W_t\}$, a newly defined $\tilde{W}_t=W_{t+r}-W_r$ where $r \geq 0$ is a Brownian motion.
I am stuck with showing it is a Gaussian process. So, for some $s<t$, I need $\tilde{W}_t-\tilde{W}_s \sim N(0,t-s)$. I simply substituted $\{W_t\}$ to end up with $W_{t+r}-W_{s+r}$. I managed showing the mean $\mu=0$ for this random variable.
I am confused with the variance. I think I am doing some very fundamental error but I don't see where. So I need $\text{Var}[W_{t+r}-W_{s+r}]$. I understand the definition of variance to be, $\mathbb{E}[X^2]-\mu^2$. Now I showed that $\mu=0$ for our random variable so essentially, I need to compute,
$$\mathbb{E}[(W_{t+r}-W_{s+r})^2]$$
By expansion, I obtain $\mathbb{E}[W_{t+r}^2-2W_{t+r}W_{s+r}+W_{s+r}^2]$ and by linearity, $$\mathbb{E}[W_{t+r}^2]-2\mathbb{E}[W_{t+r}W_{s+r}]+\mathbb{E}[W_{s+r}^2]$$
By independence, the middle term disappears and I am left with $\mathbb{E}[W_{t+r}^2]+\mathbb{E}[W_{s+r}^2]$. Since $W_t$ is a Brownian Motion, the expression I have is simply the sum of each variance, so $\mathbb{E}[W_{t+r}^2]+\mathbb{E}[W_{s+r}^2]=(t+r)+(s+r)$. Which clearly doesn't give me $t-s$.
Can anyone please tell me what mistake I am making here? Is it something arithmetic? Error in the use of variance and mean? I need to know why I am wrong...Thank you
That's not correct - note that $W_{t+r}$ and $W_{s+r}$ are not independent. The increments of Brownian motion are independent, i.e. $W_{t+r}-W_{s+r}$ and $W_{s+r}-W_0 = W_{s+r}$ are independent for $s<t$. Therefore,
$$\mathbb{E}(W_{t+r} W_{s+r}) = \mathbb{E}((W_{t+r}-W_{s+r}) W_{s+r}) + \mathbb{E}(W_{s+r}^2) = 0 + (s+r),$$
and this gives
$$\mathbb{E}((W_{t+r}-W_{s+r})^2) = (t+r) - 2(s+r) + (s+r) = t-s.$$
Finally, let me remark that it is much easier to calculate the variance using the stationarity of the increments of the (original) Brownian motion $(W_t)_{t \geq 0}$: Since $W_u - W_v \sim W_{u-v}$ for any $u \geq v$, we have $W_{t+r}-W_{s+r} \sim W_{t-s}$; in particular
$$\mathbb{E}((W_{t+r}-W_{s+r})^2) = \mathbb{E}(W_{t-s}^2) = t-s.$$