Electronic Communications in Probability

Cramér theorem for Gamma random variables

Solesne Bourguin and Ciprian Tudor

Full-text: Open access


In this paper we discuss the following problem: given a random variable $Z=X+Y$ with Gamma law such that $X$ and $Y$ are independent, we want to understand if then $X$ and $Y$ each follow a Gamma law. This is related to Cramer's theorem which states that if $X$ and $Y$ are independent then $Z=X+Y$ follows a Gaussian law if and only if $X$ and $Y$ follow a Gaussian law. We prove that Cramer's theorem is true in the Gamma context for random variables living in a Wiener chaos of fixed order but the result is not true in general. We also give an asymptotic variant of our result.

Article information

Electron. Commun. Probab. Volume 16 (2011), paper no. 34, 365-378.

Accepted: 7 July 2011
First available in Project Euclid: 7 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F05: Central limit and other weak theorems
Secondary: 60H05: Stochastic integrals 91G70: Statistical methods, econometrics

Cramér's theorem Gamma distribution multiple stochastic integrals limit theorems Malliavin calculus

This work is licensed under a Creative Commons Attribution 3.0 License.


Bourguin, Solesne; Tudor, Ciprian. Cramér theorem for Gamma random variables. Electron. Commun. Probab. 16 (2011), paper no. 34, 365--378. doi:10.1214/ECP.v16-1639. http://projecteuclid.org/euclid.ecp/1465261990.

Export citation


  • H. Cramér. Uber eine Eigenschaft der normalen Verteilungsfunction. Math. Z. 41(2) (1936), 405-414.
  • Y. Hu and D. Nualart. Some processes associated with fractional Bessel processes. Journal of Theoretical Probability 18(2) (2005), 377-397.
  • P. Malliavin. Stochastic Analysis. Springer-Verlag, Berlin (1997).
  • D. Nualart. Malliavin Calculus and Related Topics. Springer-Verlag, Berlin (2006).
  • I. Nourdin and G. Peccati. Noncentral convergence of multiple integrals. The Annals of Probability 37(4) (2009), 1412-1426.
  • I. Nourdin and G. Peccati. Stein's method on Wiener chaos. Probability Theory and Related Fields 145(1-2) (2007), 75-118.
  • C.A. Tudor. Asymptotic Cramér's theorem and analysis on Wiener space. To appear in Séminaire de Probabilités (2008). number not available.
  • A.S. Ustunel and M. Zakai On independence and conditioning on Wiener space. The Annals of Probability 17(4) (1989), 1441-1453.