The one cross term the fourth power of a sum does not kill

Recorded description

Let Y be a family of rational-valued random variables on a finite index range [0, n) with weights p, and let z be any further rational-valued variable on the same range. If for every pair of distinct indices i, i' below m the mixed expectation E[Y_i Y_i' z^2] vanishes, then the expectation of the square of the pointwise sum of the first m, times z^2, is the sum over i < m of E[Y_i^2 z^2]. The off-diagonal pairs die and the diagonal survives; that surviving diagonal is the only term the multinomial expansion of a fourth power does not kill under four-wise uncorrelatedness. The pairing hypothesis is spelled out rather than packaged as Rat.FourwiseUncorrelated because the induction runs over m while z is a variable OUTSIDE that range -- in the intended use it is Y_m itself -- so the hypothesis must be uniform in m, and this one is.