Journal of Theoretical Probability

, Volume 27, Issue 4, pp 1059–1070 | Cite as

Representations of the Absolute Value Function and Applications in Gaussian Estimates



We study the expectation of unsigned Gaussian quadratic forms and negative absolute moments of Gaussian products. The main tool we use is the integral representation of the absolute value function.


Gaussian estimates Integral representations Quadratic form Negative moments 

Mathematics Subject Classification (2010)

Primary 60E15 Secondary 62H12 


  1. 1.
    Ball, K.M.: The complex plank problem. Bull. Lond. Math. Soc. 33, 433–442 (2001)CrossRefMATHGoogle Scholar
  2. 2.
    Benítez, C., Sarantopoulos, Y., Tonge, A.: Lower bounds for norms of products of polynomial. Math. Proc. Camb. Philos. Soc. 124, 395–408 (1998)CrossRefMATHGoogle Scholar
  3. 3.
    Frenkel, P.E.: Pfaffians, hafnians and products of real linear functionals. Math. Res. Lett 15(2), 351–358 (2008)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)CrossRefMATHGoogle Scholar
  5. 5.
    Karlin, S., Rinott, Y.: Total positivity properties of absolute value multinormal variables with applications to confidence interval estimates and related probabilistic inequalities. Ann. Stat. 9(5), 1035–1049 (1981)Google Scholar
  6. 6.
    Klartag, B., Vershynin, R.: Small ball probability and Dvoretzky theorem. Israel J. Math. 157(1), 193–207 (2007)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Logan Jr, B.F., Mazo, J.E., Odlyzko, A.M., Shepp, L.A.: On the average product of Gauss-Markov variables. Bell Syst. Tech. J. 62(10), 2993–3003 (1983)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Li, W.V., Wei, A.: On the expected number of zeros of random harmonic polynomials. Proc. Am. Math. Soc. 137, 195–204 (2009)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Li, W.V., Wei, A.: Gaussian integrals involving absolute value functions. In: IMS collections, High Dimensional Probability V: The Luminy Volume, vol. 5, pp. 43–59 (2009)Google Scholar
  10. 10.
    Li, W.V., Wei, A.: A Gaussian inequality for expected absolute products. J. Theor. Probab (2010)Google Scholar
  11. 11.
    Li, W.V., Wei, A.: Wick formulas for quaternion Gaussian and \(\beta \)-permanental variables. In: Zhang, T., Zhou, X. (eds.) Stochastic Analysis and Its Applications. Series of Interdisciplinary Mathematical Sciences, pp. 291–300 (2012)Google Scholar
  12. 12.
    McLennan, A.: The expected number of real roots of a multihomogeneous system of polynomial equations. Am. J. Math. 124(1), 49–73 (2002)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Nabeya, S.: Absolute moments in 3-dimensional normal distribution. Ann. Inst. Stat. Math. Tokyo 4, 15–30 (1952)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Vere-Jones, D.: Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. N. Z. J. Maths. 26, 125–149 (1997)MATHMathSciNetGoogle Scholar
  15. 15.
    Vitale, R.A.: On the Gaussian representation of intrinsic volumes. Stat. Probab. Lett. 78(10), 1246–1249 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.University of RochesterRochesterUSA

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