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 \(\left(X_1,\ldots,X_n\right)\) determine the value of \(\E\left(X_1+\cdots+X_n\right)\), whenever this expectation is defined.

For every \(n\ge2\), we prove that this marginal degree is \(\left\lceil n/2\right\rceil\).

The upper bound follows from a theorem of Simons (1977).

The lower bound is proved by constructing, for every \(1\le k<\left\lceil n/2\right\rceil\), 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.