Exchangeable Random Variables

Let $\left\{ X_{i}: 1 \leq i \leq n+1\right\}$ be a finite collection of random variables such that conditional $X_{n+1},\left\{ X_{1}, X_{2}, \ldots , X_{n}\right\}$ are iid. Show that $\left\{ X_{i}: 1 \leq i \leq n\right\}$ is exchangeable.

Let $\left\{ X_{i}: 1 \leq i \leq n\right\}$ be exchangeable. Suppose $\mathbb {E}\left[\left|X_{1}\right|\right]< \infty$. Show that

Let $\left\{ X_{i}\right\}_{i \geq 1}$ be random variables on a probability space $(\Omega , \mathcal{F}, P)$. Suppose for some $\sigma$-algebra $\mathcal{G} \subset \mathcal{F}$ (4.7) holds. Show that $\left\{ X_{i}\right\}_{i \geq 1}$ are exchangeable.

An urn contains initially one red ball and one green ball. At later stages, a ball is picked from the urn uniformly at random, and is returned to the urn together with a fresh ball of the same colour. Assume the usual independence. Let $X_{k}$ be the indicator function of the event that the $k$ th ball picked is red.

In a game of dice one wishes to use one of two dice $A$ and $B$. $A$ has two white and four red faces and $B$ has two red and four white faces. A coin is tossed in order to decide which die is to be used and that die is then used throughout. Let $\left\{ X_k, k \geq 1\right\}$ be a sequence of random variables defined as follows:

at the $k$th roll of the die. Show that the law of large numbers does not hold for the sequence $\left\{ X_k, k \geq 1\right\}$. Why is this the case?

Let $(X, \mathcal{A}, P)$ be a probability space, and let $\mathbb {P}$ equal $P^{n}$, the $n$-fold product measure on $\mathcal{A}^{\mathbb {N}}$. For each $x$ in $X$, let $\delta (x)$ denote the point mass at $x$. For $\mathbf{x}$ in $X^{n}$, let $T(\mathbf{x})$ denote the so-called empirical measure, $n^{-1} \sum_{i \leq n} \delta \left(x_{i}\right)$. Intuitively, if we are given the measure $T(\mathbf{x})$, the conditional distribution for $\mathbb {P}$ should give mass $1 / n$ ! to each of the $n$ ! permutations $\left(x_{1}, \ldots , x_{n}\right)$. Formalize this intuition by constructing a ($T, T \mathbb {P}$)-disintegration for $\mathbb {P}$. Warning: Not as easy as it seems. What is the $(\mathcal{T}, \mathcal{B})$ for this problem? What happens if the empirical measure is not supported by $n$ distinct points? How should you define $\mathbb {P}_{t}$ when $t$ does not correspond to a possible realization of an empirical measure?

Suppose $\left\{ X_{n}: n \in \mathbb {N}\right\}$ is an exchangeable sequence of square-integrable random variables. Show that $\operatorname {cov}\left(X_{1}, X_{2}\right) \geq 0$.

(Hewitt Savage 1955, Section 5) Let $\mathbb {P}$ be exchangeable, in the sense of Definition $\langle 49 \rangle$.

Definition 49: A probability measure $\mathbb {P}$ on $\mathcal{A}^{\mathbb {N}}$ is exchangeable if it is invariant under $S_{\pi }$ for every finite permutation $\pi$, that is, if $\mathbb {E}\left[h\right] = \mathbb {E}\left[h_{\pi }\right]$ for every $h$ in $\mathcal{M}^{+}\left(X^{\mathbb {N}}, \mathcal{A}^{\mathbb {N}}\right)$ and every finite permutation $\pi$. Equivalently, under $\mathbb {P}$ the random vector $\left(x_{\pi (1)}, x_{\pi (2)}, \ldots , x_{\pi (n)}\right)$ has the same distribution as $\left(x_{1}, x_{2}, \ldots , x_{n}\right)$, for every $n$-permutation $\pi$, and every $n$.