Is $X_t := W_t^2$ a Wiener process for a Wiener process $(W_t)_{t \geq 0}$?

Question

I'm studying for exam and found this exercise which I don't really understand:

Suppose $W_t$ is standard Wiener process. Is process $X_t=W_t^2, t\geq0$ a Wiener process?

So I need to show that $W_t^2$ is Gaussian and has properties of Wiener process?

Answer 1

For any Wiener process $(B_t)_t$, we have $B_t \sim N(0,t)$; in particular $\mathbb{P}(B_t < 0) = \frac{1}{2}$. Since $X_t= W_t^2 \geq 0$ is non-negative, we have $\mathbb{P}(X_t<0)=0$ and therefore $(X_t)_{t \geq 0}$ cannot be a Wiener process.