On measures of non-degeneracy

Abstract

If φ is a convex function and X a random variable then (by Jensen's inequality) ψ _φ (X) = E_φ (X) – φ (EX) is non-negative and 0 iff either φ is linear in the range of X or X is degenerate. So if φ is not linear then ψ _φ (X) is a measure of non-degeneracy of the random variable X. For φ (x) = x², ψ _φ (X) is simply the variance V(X) which is additive in the sense that V(X + Y) = V(X) + V(Y) if X and Y are uncorrelated. In this note it is shown that if φ ″(·) is monotone non-increasing then ψ _φ is sub-additive for all (X, Y) such that EX ≧ 0, P(Y ≧ 0) = 1 and E(X | Y) = EX w.p.l, and is additive essentially only if φ is quadratic. Thus, it confirms the unique role of variance as a measure of non-degeneracy. An application to branching processes is also given.