What to Expect When You're Expecting

The marginal degree of sums in dimension $n$ is the smallest integer $k$ such that the joint distributions of all subcollections of at most $k$ coordinates of a real-valued random vector $(X_1,\dots ,X_n)$ determine the value of $\text{E}(X_1+\cdots +X_n)$, whenever this expectation is defined. For every $n\ge 2$, we prove that this marginal degree is $\lceil n\mathrm{/}2\rceil$. The upper bound follows from a theorem of Simons (1977). The lower bound is proved by constructing, for every $1\le k<\lceil n\mathrm{/}2\rceil$, two joint laws whose marginals of dimension at most $k$ agree, but for which the corresponding expectations of $X_1+\cdots +X_n$ are defined and unequal.