exchangeability vs. shift invariance

Question

Let $X_n$ be a sequence of real RVs indexed by $n \geq 0$. Can someone provide an example of a sequence $X_n$ that is exchangeable (law invariant under finitely supported permutations on the natural numbers) but not shift invariant (law invariant under left shift)? What about the other way?

Answer 1

If the sequence $(X_n)$ is exchangeable then $(X_1,X_2,\ldots,X_n,X_{n+1})$ and $(X_2,\ldots,X_n,X_{n+1},X_1)$ coincide in distribution, in particular $(X_1,X_2,\ldots,X_{n})$ and $(X_2,\ldots,X_{n+1})$ coincide in distribution. This holds for every $n\geqslant1$ hence $(X_n)$ is shift invariant. Thus, exchangeability implies shift invariance.

Let $X_0$ denote a nondegenerate symmetric random variable and $X_n=(-1)^nX_0$ for every $n$. Then $(X_n)$ is shift invariant but the distributions of $(X_1,X_2)$ and $(X_1,X_3)$ are different hence $(X_n)$ is not exchangeable. Thus, shift invariance does not imply exchangeability.