Let $X$ be a Polish space. Let $(\mu_n)_{n \in \mathbb{N} \cup \{\infty\}}$ be a family of Borel probability measures $\mu_n$ on $X$ such that $\mu_n \to \mu_\infty$ weakly as $n \to \infty$. For each $k \in \mathbb{N} \cup \{\infty\}$, define the projection
\begin{align*}
\pi_k \colon X^{\mathbb{N} \cup \{\infty\}} \to X& \\
\pi_k\!\!\left( (x_n)_{n \in \mathbb{N} \cup \{\infty\}} \right) = x_k&
\end{align*}
Does there necessarily exist a Borel probability measure $\mu$ on the product space $X^{\mathbb{N} \cup \{\infty\}}$ such that
- for each $n \in \mathbb{N} \cup \{\infty\}$, the pushforward of $\mu$ under $\pi_n$ is $\mu_n$;
- $\pi_n$ converges $\mu$-almost surely to $\pi_\infty$ as $n \to \infty$?
If so, is there a name for this result? Or a standard reference?
[I seem to recall reading somewhere that there necessarily exists $\mu$ such that the first property holds and the second holds with convergence in probability; so the point of my question is whether this convergence in probability can be strengthened to almost-sure convergence.]
Best Answer
Skorokhod's representation theorem gives representation $X_{n}\to X$ of almost sure convergence and here is a proof for it.