Which distributions have the Matsumoto-Yor property?

Angelo Efoévi Koudou (1) , Pierre Vallois (2, 1)

1 IECN - Institut Élie Cartan de Nancy
2 BIGS - Biology, genetics and statistics

For four types of functions ξ : ]0, ∞[→]0, ∞[, we characterize the law of two independent and positive r.v.'s X and Y such that U := ξ(X + Y) and V := ξ(X) − ξ(X + Y) are independent. The case ξ(x) = 1/x has been treated by Letac and Weso lowski (2000). As for the three other cases, under the weak assumption that X and Y have density functions whose logarithm is locally integrable, we prove that the distribution of (X, Y) is unique. This leads to Kummer, gamma and beta distributions. This improves the result obtained in [1] where more regularity was required from the densities.

Mots clés: Kummer distribution; Gamma distribution; generalized inverse Gaussian distribution; Matsumoto-Yor property; Beta distribution

Domaines: Probabilités [math.PR]; Statistiques [math.ST]

Angelo Efoévi Koudou, Pierre Vallois. Which distributions have the Matsumoto-Yor property?. Electronic Communications in Probability, 2011, 16, pp.49. ⟨10.1214/ECP.v16-1663⟩. ⟨hal-01283962⟩