Let $X_1,\dots,X_n$ be i.i.d, with $X_1$ having a continuous cumulative distribution function. Let $T = (X_{(1)},\dots,X_{(n)})$ be the order statistics. Show given $T$, the conditional distribution of $X=(X_1,\dots,X_n)$ is a discrete distribution putting probability $1/n!$ on each of the $n!$ points $(X_{\pi(1)},\dots,X_{\pi(n)})$ for $\pi \in S_n$ the symmetric group. As I'm not comfortable with conditional distributions, I'd like some feedback on my attempt.
My thoughts on this problem: I think basically we are working over the canonical probability space, i.e. $X_1: (\mathbb R,\mathcal B(\mathbb R)) \to (\mathbb R,\mathcal B(\mathbb R))$ is the identity map, and $F:t\mapsto P(X_1\leq t)$ is continuous. There is a regular conditional distribution $\mu:\mathcal B(\mathbb R^n) \times \mathbb R^n \to [0,1]$ such that
- For each $x=(x_1,\dots,x_n)\in \mathbb R^n$, $\mu(\cdot,x)$ is a probability measure on $(\mathbb R^n, \mathcal B(\mathbb R^n))$.
- For each $B\in \mathcal B(\mathbb R^n), \mu(B,\cdot)$ is a version of $P(X \in B \mid T)$
Hence we want to show for each $x\in \mathbb R^n$ and show that $\mu(\cdot,x)$ agrees with the claim, that is, $$ \mu(\{\pi(x)\},x) = 1/n!, \quad \pi(x)\overset{\text{def}}{=}(x_{\pi(1)},\dots,x_{\pi(n)}), \pi \in S_n$$ Now, given $B\in \mathcal B(\mathbb R^n)$, set $$B_{\pi}= B\cap \{x: x_{\pi(1)}\leq x_{\pi(2)}\leq\dots\leq x_{\pi(n)}\}$$ We would like to compute $$P(X\in B, T\in G) = \int_G \mu(B,x)P(dx) = \sum_{\pi\in S_n}\int_G \mu(B_{\pi},x)P(dx), \quad G\in \mathcal B(\mathbb R^n), G\subset \{x_1<\dots<x_n\}$$ Assuming that $B$ is symmetric in the sense that $x\in B$ implies $\pi(x)\in B$ for all $\pi \in S_n$, and $G\subset B$ with $P(T\in G)>0$, then the above implies $P(T\in G, X\in B_{\pi})= P(T\in G)/n!$ This is true if we can justify $\mu(B_{\pi},x) = \mu(B_{\tau},x)$ for every $\tau \in S_n$ and $x\in G$, this should follow from the i.i.d assumption, but I don't know how to get it rigorously.
My questions:
- Any false assertions?
- How to get that $\mu(B_{\pi},x) = \mu(B_{\tau},x), \quad \tau \in S_n, x\in \mathbb R^k$?
- How to conclude that $\mu(\{\pi(x)\},x) = 1/n!$? (To be honest, this doesn't even seem like the right track, even if nothing wrong is written!)