Geodesic
Bernstein theorems and transformations of correlation measures in statistical physics
Matematičeskie zametki, Tome 79 (2006) no. 5, pp. 700-716.

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We study the class of endomorphisms of the cone of correlation functions generated by probability measures. We consider algebraic properties of the products $(\,\cdot\,{,}\,\star)$ and the maps $K$, $K^{-1}$ which establish relationships between the properties of functions on the configuration space and the properties of the corresponding operators (matrices with Boolean indices): $F(\gamma)\to \widehat F_\cup(\gamma)=\{F(\alpha\cup\beta)\}_{\alpha,\beta\subset\gamma}$. For the operators $\widehat F_\cup(\gamma)$ and $\widehat F_\cap(\gamma)$, we prove conditions which ensure that these operators are positive definite; the conditions are given in terms of complete or absolute monotonicity properties of the function $F(\gamma)$. 
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Yu. G. Kondrat'ev; A. M. Chebotarev. Bernstein theorems and transformations of correlation measures in statistical physics. Matematičeskie zametki, Tome 79 (2006) no. 5, pp. 700-716. http://geodesic.mathdoc.fr/item/MZM_2006_79_5_a6/

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