Prove that exists a sequence $(X_n)$ of i.i.d. random variables with standard normal distribution

Question

How can I prove that exist a probability space $(\Omega,\Sigma,\mathbb{P})$ and a sequence $(X_n:\Omega \to \mathbb{R})_{n\in\mathbb{N}}$ of i.i.d. random variables with standard normal distribution?

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I just know to prove the finite case: given $n\in\mathbb{N}$, there're a probability space $(\mathbb{R}^n,\mathfrak{B},\mathbb{P})$ and independent random variables $X_1,\cdots,X_n:\mathbb{R}^n\to \mathbb{R}$ such that $X_i\sim N(0,1)$ for all $i=0,\cdots,n$.

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Thank you for your attention!

Answer 1

Here is a fun constructive approach. Write $\mathbb{N} = \{1, 2, 3, \ldots\}$ for the set of all positive integers.

1. Let $(\Omega, \mathcal{F}, \mathbf{P})$ be a probability space which is ample enough to simulate a random variable $U$ having the uniform distribution over $[0, 1]$.

2. Define $A_n$, $n \in \mathbb{N}$, by $$ A_n = \lfloor 2^n U \rfloor \text{ mod } 2 \quad \in \{0, 1\}. $$ Then for any $n \in \mathbb{N}$ and for any $a_1, \ldots, a_n \in \{0,1\}$, it is easy to show that $$ \mathbf{P}(A_1 = a_1, \ldots, A_n = a_n) = 2^{-n}, $$ showing that $(A_n)_{n\in\mathbb{N}}$ are mutually independent.

   (Intuitively,. $A_n$ is the $n$th digit in the binary expansion of $U$. In particular, $U=\sum_{n=1}^{\infty} \frac{1}{2^n} A_n$ holds almost surely.)

3. Since $\mathbb{N}\times\mathbb{N}$ has the same cardinality as $\mathbb{N}$, we can find an injection $\phi : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$. Now define $(X_n)_{n\in\mathbb{N}}$ by $$ X_n = \sum_{k=1}^{\infty} \frac{1}{2^k}A_{\phi(n, k)} $$ By the construction, $X_n$'s are mutually independent. Moreover, it is not hard to show that each $X_n$ has uniform distribution over $[0, 1]$.

4. Finally, let $\Phi^{-1}$ be the inverse CDF of the standard normal distribution, and let $Z_n = \Phi^{-1}(X_n)$. Then each $Z_n$ has standard normal distribution, and $Z_n$'s are mutually independent.

Answer 2

You can use the Ionescu-Tulcea Theorem (https://en.wikipedia.org/wiki/Ionescu-Tulcea_theorem). Here the transition kernels are just $\kappa_i(x, A) = P(X_1 \in A)$.