Abstract
We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities.
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Chen, L.H.Y. (1985). Poincaré-type inequalities via stochastic integralsZeitschr. Wahrsch. verw. Geb.,69, 251–277.
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.
Crouzeix, J.P. and Ferland, J.A. (1982). Criteria for quasi-convex and pseudo-convexity: relationships and comparisons.Math. Programm. 23, 193–205.
Glimm, J. and Jaffe, A. (1981).Quantum Physics: A Functional Integral Point of View. Springer-Verlag, New York.
Houdré, C. and Kagan, A. (1995). Variance inequalities for functions of Gaussian variables.J. Th. Probab.,8, 23–30.
Houdré, C. and Pérez-Abreu. V. (1995). Covariance identities and inequalities for functionals on Wiener and Poisson spaces.Ann. Probab.,23, 400–419.
Hu, Y. (1997). Itô-Wiener chaos expansion with exact residual and correlation, variance inequalities.J. Theoret. Probab. 10, 835–848.
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.
Koldobsky, A.L. and Montgomery-Smith, S.J. (1996). Inequalities of correlation type for symmetric stable random vectors,Statist. Probab. Letters,28, 91–96.
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.
Lee, M.L.T., Rachev, S.T., and Samorodnitsky, G. (1990). Association of stable random variables.Ann. Probab.,18, 1759–1764.
Lifshits, M.A. (1995).Gaussian Random Functions. Kluwer Academic Publishers, London.
Linde, W. (1986).Probability in Banach Spaces: Stable and Infinitely Divisible Distributions. John Wiley & Sons, New York.
Martos, B. (1969). Subdefinite matrices and quadratic forms.SIAM J. Appl. Math.,17, 1215–1223.
Piterbarg, V.I. (1996).Asymptotic Methods in the Theory of Gaussian Processes and Fields. American Mathematical Society. Providence, RI.
Pitt, L.D. (1977). A Gaussian correlation inequality for symmetric convex sets.Ann. Probab.,5 470–474.
Pitt, L.D. (1982). Positively correlated normal variables are associated.Ann. Probab.,10, 496–499.
Prekopa, A. (1973). On logarithmic concave measures and functions.Acta. Sci. Math.,34, 335–343.
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.
Samorodnitsky, G. (1995). Association of infinitely divisible random vectors.Arch. Proc. Appl.,55, 45–55
Schechtman G., Schlumprecht, Th., and Zinn, J. (1998). On the Gaussian measure of the intersection of symmetric convex sets.Ann. Probab. 26, 346–357.
Szarek, S.J. and Wermer, E. (1996). A correlation inequality for the Gaussian measure. Preprint.
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.
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Communicated by Abram Kagan
Acknowledgements and Notes. The research of C. Houdré was supported in part by an NSF Mathematical Sciences Post-Doctoral Fellowship and by an NSF-NATO Postdoctoral Fellowship and by the NSF grant No. DMS-98032039. This research was completed while V. Pérez-Abreu was visiting the Georgia Institute of Technology.
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Houdré, C., Pérez-Abreu, V. & Surgailis, D. Interpolation, correlation identities, and inequalities for infinitely divisible variables. The Journal of Fourier Analysis and Applications 4, 651–668 (1998). https://doi.org/10.1007/BF02479672
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DOI: https://doi.org/10.1007/BF02479672