Let $X_t = tW_{\frac{1}{t}} \forall t>0$ and $X_0 = 0$. I am trying to show that this process is a brownian motion under some measure P.
I have shown that it is continuous and that it is distributed normally with mean zero and variance t. However, I am trying to show that it has normally distributed increments:
let $t>s\ge0$ then I want to show that $X_{t+s}-X_s \sim N(0,t)$. Showing that this has mean zero and variance t is easy. However, the solution I have states that since this is a sum of gaussian random variables, then it must also be gaussian. However, when looking online I can see that the sum of two normal rv's need not be gaussian, that is, those rvs must be jointly gaussian, which then implies that the sum is gaussian. Am I missing something, or is the solution skipping over an important distinction?