A class of random polynomials with an invariance property (Q1372405)

Let \(k\geq 1\) be a fixed integer and let \(S\) be the set of random variables \(X\) with \(EX^{2k}<\infty\) whose distribution function \(F_x\) has more than \(k\) points of increase. For fixed \(X\in S\) let \(\widehat E_k[Y|X]\) be the projection of \(Y\) on the linear span of \(1,X,\dots, X^k\), with projection defined by the inproduct \((U,V)= EUV\). The authors define functions of the following form: \[ Q(x)\equiv Q(X, F_x)= \sum^k_{i=0} a_i(EX,\dots, EX^{2k})X^k, \] where \(a_0,\dots, a_k\) are fixed functions \(\mathbb{R}^{2k}\to\mathbb{R}\). The authors characterize the class of those \(Q\) that satisfy \(\widehat E_k[Q(X)|X+Y]= Q(X+ Y)\) for all pairs of independent \(X\) and \(Y\) in \(S\), and show that then \[ \{\text{Var }Q(X+ Y)\}^{-1}\geq \{\text{Var }Q(X)\}^{-1}+ \{\text{Var }Q(Y)\}^{-1}. \] A special case is \(Q(X, F_X)=\widehat E_k(f'(X)/f(X)|X)\) when \(X\) has density \(f\). There is a connection with Fisher's information.