# zbMATH — the first resource for mathematics

Interpolation, correlation identities, and inequalities for infinitely divisible variables. (English) Zbl 0924.60006
The authors derive formulas for expressions of the form $$\mathbb{E} G(X_1)- \mathbb{E} G(X_0)$$ as a (kind of) convex combination of $$\mathbb{E} G(X_\alpha)$$, where the random vector $$X_\alpha$$ is a (kind of) convex combination of random vectors $$X_1$$ and $$X_0$$. A frequently used example concerns $$X_\alpha$$ with characteristic function $$\varphi_1^\alpha \varphi_0^{1-\alpha}$$, where the characteristic functions $$\varphi_1$$ and $$\varphi_0$$ of $$X_1$$ and $$X_0$$ are infinitely divisible. The simplest case concerns a known formula for Gaussian random vectors. Special cases yield results for variances and covariances; the latter leading to results on association. Repeated application of these formulas gives rise to expansions for variances and inequalities for covariances. The paper concludes with some ‘evidence’ in favour of the so-called Gaussian correlation conjecture: $$\text{Cov(\textbf{1}}(X\in A_1), \mathbf{1}(X\in A_2))\geq 0$$, where $$X$$ is a $$d$$-dimensional Gaussian random vector and $$A_1$$ and $$A_2$$ are symmetric convex sets in $$\mathbb{R}^d$$, and $$d\geq 3$$.

##### MSC:
 60E07 Infinitely divisible distributions; stable distributions 60E10 Characteristic functions; other transforms 60E15 Inequalities; stochastic orderings 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Full Text:
##### References:
 [1] Chen, L.H.Y. (1985). Poincaré-type inequalities via stochastic integralsZeitschr. Wahrsch. verw. Geb.,69, 251–277. · Zbl 0549.60019 · doi:10.1007/BF02450283 [2] Chen, L.H.Y. and Lou, J.H. (1987). Characterization of probability distributions by Poincaré-type inequalities.Ann. Inst. H. Poincaré, Sec B,23, 91–110. · Zbl 0612.60013 [3] Crouzeix, J.P. and Ferland, J.A. (1982). Criteria for quasi-convex and pseudo-convexity: relationships and comparisons.Math. Programm. 23, 193–205. · Zbl 0479.90067 · doi:10.1007/BF01583788 [4] Glimm, J. and Jaffe, A. (1981).Quantum Physics: A Functional Integral Point of View. Springer-Verlag, New York. · Zbl 0461.46051 [5] Houdré, C. and Kagan, A. (1995). Variance inequalities for functions of Gaussian variables.J. Th. Probab.,8, 23–30. · Zbl 0815.60018 · doi:10.1007/BF02213451 [6] Houdré, C. and Pérez-Abreu. V. (1995). Covariance identities and inequalities for functionals on Wiener and Poisson spaces.Ann. Probab.,23, 400–419. · Zbl 0831.60029 · doi:10.1214/aop/1176988392 [7] Hu, Y. (1997). Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities.J. Theoret. Probab. 10, 835–848. · Zbl 0891.60060 · doi:10.1023/A:1022654314791 [8] Karlin, S. (1994). A general class of variance inequalities. In:Multivariate Analysis: Future Directions. Rao, C.R. and Patil, G.P., Eds., North Holland, Amsterdam, 279–294. [9] Koldobsky, A.L. and Montgomery-Smith, S.J. (1996). Inequalities of correlation type for symmetric stable random vectors,Statist. Probab. Letters,28, 91–96. · Zbl 0855.60016 · doi:10.1016/0167-7152(95)00096-8 [10] Ledoux, M. (1995). L’algèbre de Lie des gradients itérés d’un générateur Markovien-développemenets de moyennes et entropies.Ann. Scient. Éc. Norm. Sup.,28, 435–460. · Zbl 0842.60075 [11] Lee, M.L.T., Rachev, S.T., and Samorodnitsky, G. (1990). Association of stable random variables.Ann. Probab.,18, 1759–1764. · Zbl 0716.60017 · doi:10.1214/aop/1176990646 [12] Lifshits, M.A. (1995).Gaussian Random Functions. Kluwer Academic Publishers, London. · Zbl 0832.60002 [13] Linde, W. (1986).Probability in Banach Spaces: Stable and Infinitely Divisible Distributions. John Wiley & Sons, New York. · Zbl 0665.60005 [14] Martos, B. (1969). Subdefinite matrices and quadratic forms.SIAM J. Appl. Math.,17, 1215–1223. · Zbl 0186.34201 · doi:10.1137/0117112 [15] Piterbarg, V.I. (1996).Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society. Providence, RI. · Zbl 0841.60024 [16] Pitt, L.D. (1977). A Gaussian correlation inequality for symmetric convex sets.Ann. Probab.,5 470–474. · Zbl 0359.60018 · doi:10.1214/aop/1176995808 [17] Pitt, L.D. (1982). Positively correlated normal variables are associated.Ann. Probab.,10, 496–499. · Zbl 0482.62046 · doi:10.1214/aop/1176993872 [18] Prekopa, A. (1973). On logarithmic concave measures and functions.Acta. Sci. Math.,34, 335–343. · Zbl 0264.90038 [19] Resnick, S.I., (1988). Association and multivariate extreme value distributions. In:Studies in Statistical Modeling and Statistical Science, Heyde, C.C., Ed., Statistical Society of Australia. · Zbl 0672.62066 [20] Samorodnitsky, G. (1995). Association of infinitely divisible random vectors.Arch. Proc. Appl.,55, 45–55 · Zbl 0817.60009 [21] Schechtman G., Schlumprecht, Th., and Zinn, J. (1998). On the Gaussian measure of the intersection of symmetric convex sets.Ann. Probab. 26, 346–357. · Zbl 0936.60015 · doi:10.1214/aop/1022855422 [22] Szarek, S.J. and Wermer, E. (1996). A correlation inequality for the Gaussian measure. Preprint. [23] Vitale, R. (1989). A differential version of the Efron-Stein inequality: Bounding the variance of a function of an infinitely divisible variable.Statist. Probab. Letters,7, 105–112. · Zbl 0664.60025 · doi:10.1016/0167-7152(88)90034-X
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.