My classmate and I are disagreeing about a point about $X(t)$, a BM.
His statement is that $X(t)$ is normal, my statement is that only increments of $X(t)$ are normal (thus (thus $X(t) - X(0)$ is normal, but can't say $X(t)$ is normal if $X(0)$ not equal to $0$).
His argument is that since $X(t) - X(0)$ is normal, then a normal plus a constant is normal, so $X(t) - X(0) + X(0) = X(t)$ must be normal.
My argument is that $X(t)$ is not normal generally because $X(0)$ might not be $0$ generally and thus $X(t)$ is not an increment (which increments of BM are guaranteed to be normal).
Who is correct?
The usual convention is that $X(0)$ is deterministic. In that case $X(t)$ is normal, with mean $X(0)$ and variance $t$. If you allow $X(0)$ to be random and not normally distributed, then $X(t)$ will not be normal.
https://www.yuval-peres-books.com/brownian-motion/