## Electronic Communications in Probability

### Cramér theorem for Gamma random variables

#### Abstract

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

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

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

https://projecteuclid.org/euclid.ecp/1465261990

Digital Object Identifier
doi:10.1214/ECP.v16-1639

Mathematical Reviews number (MathSciNet)
MR2819659

Zentralblatt MATH identifier
1272.68116

Rights

#### Citation

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. https://projecteuclid.org/euclid.ecp/1465261990

#### References

• 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.