Hi everyone I have the following problem. I'd appreciate any help.
Let $\{Y_n:n\in \mathbb{N}\}$ be a sequence of real valued random variables on a probabilility space $(\Omega,\mathscr{F}, P)$. Let $n\ge 1$ a fixed integer. Define $Z_n=(Y_1,\ldots,Y_n)$ and $W_n=(Y_{n+1},\ldots)$.
i) Show that $W_n$ and $Z_n$ are independent.
ii) Assume furthermore that the random variables $Y_n$ are identically distributed, that is the distribution function satisfies $F_{Y_n}=F_{Y_m}$ for every $0\le n<m<\infty$. Then prove that for every $n\ge 1$, $Z_n$ and $W_n$ have the same distribution.
For i) what I proved using Dynkin classes is that $\mathcal{T}_n =\sigma \left( \bigcup_{k=n+1}^\infty \sigma(Y_k) \right)$ and $\mathcal{B}=\sigma\left(\bigcup_{k=1}^n \sigma(Y_k) \right)$ are independent under $P$, where $\sigma(Y_k)$ is the minimum sigma algebra which makes $Y_k$ measurable. But from here I'm not sure how to conclude.
For $ii)$ I don't understand what is the distribution of an infinite dimensional random vector.
I have no experience working with infinite dimensional random vectors. I'd appreciate any help please. (If someone considers it necessary I can add the proof of the independence using Dynkin classes, but this is standard.)