A differential version of the Efron-Stein inequality: Bounding the variance of a function of an infinitely divisible variable
Let X be an infinitely divisible random variable with characteristic function in Lévy-Khinchine form: \[ \phi_ X(t)=\exp [itb-\sigma^ 2t^ 2/2+\int^{\infty}_{-\infty}(e^{itu}-1-itu/(1+u^ 2))(1+u^ 2)u^{-2} \nu (du)], \] where \(\nu\) is a finite measure with \(\nu (\{0\})=0\). If \(G(x)=\int^{x}_{x_ 0}g(u)dy+C_ 0\), then \[ var G(X)\leq \sigma^ 2E g^ 2(X)+\int^{\infty}_{-\infty}[u^{- 1}\int^{u}_{0}E g^ 2(X+y)dy](1+u^ 2)\nu (du). \] Equality occurs in the case of a finite upper bound if and only if \(G(X)=aX+b\) a.s. with some constants a and b. The result can be derived rom results of \textit{B. Efron} and \textit{C. Stein} [Ann. Stat. 9, 586-596 (1981; Zbl 0481.62035)].
scientific article; zbMATH DE number 4086641
scientific article