I know how to calculate the mean and variance, but have no idea how to prove that it is a normal distribution given that W(t) and W(t/2) are not independent.
Also, is the process M(t)=W(t)+W(t/2) a Brownian motion? It can be shown that Var[M(t)]=5/2t. not t, so it's not a Brownian motion. However, how do I show that its increments are not independent?
Thanks for any help.
Hello and welcome to math.stackexchange.
Regarding your first question: Let $X(t) = W(t/2)$ and $Y(t) = W(t) - W(t/2)$. These r.v.s are independent for any $t > 0$. You want the distribution of $2X(t) + Y(t)$. Take it from there.
Regarding your second question: Consider $M(t) = W(t) + W(t/2)$ and the increment $$ M(2t) - M(t) = W(2t) + W(t) - (W(t) + W(t/2)) = W(2t) - W(t) + M(t) \, $$ Now recall that $M(t)$ and $W(2t)-W(t)$ are surely independent and compute the covariance.