Does there exist random variables such that any strict subset of $\{ X_1, X_2, ... X_n\}$ is independent but $X_1, X_2, ... X_n$ are not independent?

Question

Let $n$ be any positive integer larger than $1$. I want to construct random variables $X_1, X_2, ... X_n$such that any strict subset of $\{ X_1, X_2, ... X_n\}$ is independent but $X_1, X_2, ... X_n$ are not independent. Any hints or reference are appreciated.

Answer 1

Let $X_1$, ... , $X_{n-1}$ be i.i.d random variable taking values in $\lbrace -1, 1 \rbrace$ uniformly and let $$X_n = \prod_{i=1}^{n-1} X_i$$ Clearly they are not independent but if you take any subset of those of size at most $n-1$, they are independent:

Let $A\subsetneq \lbrace 1, \dots , n\rbrace$, if $n\notin A$ then independence is trivial so suppose $n\in A$. In that case there is $i\in[1:n-1]$ such that $i\notin A$ but the distribution of all the variables can be rewritten, by symmetry, as $\lbrace X_j \rbrace_{j\neq i}$ are i.i.d random variable uniformly distributed in $\lbrace -1, 1 \rbrace$ and $X_i=\prod\limits_{j\neq i} X_j$ and so the subset is independent.