Independent random variables $X_1$, $X_2$, $X_3$, ... defined on $(\Omega, \mathcal{F}, \mathbb{P})$, each normal mean zero, variance $1$.

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be the unit interval with Lebesgue measure on the Borel subsets. Show that one can find independent random variables $X_1, X_2, X_3, \dots$ defined on $(\Omega, \mathcal{F}, \mathbb{P})$, each normal mean zero, variance $1$.

I know how to show the following.

• Defining$$X = .d_1d_2 \dots = \sum_{j = 1}^\infty 2^{-j}d_j$$where the $d_j$ are i.i.d. random variables with$$P(d_j = 0) = P(d_j = 1) = {1\over2},$$we have that $X$ is uniformly distributed on $[0, 1]$.
• Using a diagonalization process, we can define $U_1, U_2, \dots$ independent random variables each with a uniform distribution on $[0, 1]$.

My question is, how do I see that $X_j = \Phi^{-1}(U_j)$ has a normal distribution with mean zero, variance one, where $\Phi$ is the normal distribution function? I just need this last step to finish.

Answer 1

Because of the monotonicity and continuity of $\Phi$, we have

$$\{x; \Phi^{-1}(x) \leq c\} = \{y; y \leq \Phi(c)\}$$

for any constant $c$. This implies

$$\mathbb{P}(X_j \leq c) = \mathbb{P}(\Phi^{-1}(U_j) \leq c) = \mathbb{P}(U_j \leq \Phi(c)).$$

Now use that $U_j$ is uniformly distributed on $[0,1]$ to finish the proof.