مرکز منطقه ای اطلاع رساني علوم و فناوري - Generalized form of Price´s theorem and its converse

Abstract :

The case of $n$ unity-variance random variables $x_{1}, x_{2},\\cdots , x_{n}$ governed by the joint probability density $w(x_{1}, x_{2}, \\cdots x_{n})$ is considered, where the density depends on the (normalized) cross-covariances $\\rho_{ij} = E [(x_{i}- \\bar{x}_{i})(x_{j} - \\bar{x}_{j})]$ . It is shown that the condition $(^{\\ast }) frac{\\delta }{\\delta \\rho_{ij}}{E[f(x_{1}, x_{2}, \\cdots , x_{n})} = E|frac{\\delta ^{2}}{\\delta x_{i} \\delta x_{j}}f(x_{1}, x_{2}, \\cdots , x_{n})| mbox{(\\i\\neq j)}$ holds for an "arbitrary" function $f(x_{1}, x_{2}, \\cdots , x_{n})$ of $n$ variables if and only if the underlying density $w(x_{1}, x_{2}, \\cdots , x_{n})$ is the usual $n$ -dimensional Gaussian density for correlated random variables. This result establishes a generalized form of Price\´s theorem in which: 1) the relevant condition $(^{\\ast })$ subsumes Price\´s original condition; 2) the proof is accomplished without appeal to Laplace integral expansions; and 3) conditions referring to derivatives with respect to diagonal terms $\\rho_{ij}$ are avoided, so that the unity variance assumption can be retained.