The Matsumoto-Yor (MY) property says that when X and Y are independent GIG (generalized inverse Gaussian) and Gamma variables with suitably related parameters then U = 1/(X+Y) and V = 1/X-1/(X+Y) are also independent and have the GIG and Gamma distributions. The property was discovered originally in Matsumoto and Yor (2001) inside the conditional structure of the exponential Brownian motion. Matsumoto and Yor (2003) established also its connection with the first and the last hitting times of Brownian motions with drifts. In Letac and Wesolowski (2000) we proved a related characterization which says that if X and Y are independent, and U and V, as defined above, are also indpendent, then X and Y (and thus U and V) are GIG and Gamma with suitably related parameters.

In meantime the MY property and related characterizations were developed in many directions as: characterizations by constancy of regression of V given U instead of independence; matrix variate version of the MY property and related characterizations of the Wishart and GIG random matrices; representation through the conditional structure of Wishart matrices; tree multivariate MY property with relations to hitting times of Brownian motions and to the conditional structure of the exponential Brownian motion as well as to so-called Sabot-Tarres-Zheng integral; free probability version of the MY property and related characterizations of the Marchenko-Pastur and free-GIG distributions.

Recently, the MY property, even its generalized form, was discovered by Croydon and Sasada (2020) within the discrete Korteg-de Vries model. They noticed that when X and Y are independent GIG variables with suitably related parameters, then U = Y(aXY +1)/(bXY +1) and V = X(aXY +1)/(bXY+1) for distinct a,b>=0 are also independent GIG random variables. It is not diffcult to see that the case a = 1 and b = 0 reduces to the original MY property. Bao and Noack (2021) proved a related characterization of the GIG distribution by independence of X and Y and of U and V under technical assumptions of strictly positive twice dfferentiable densities. Very recently, in Letac and Wesolowski (2022), we have given complete proof of the characterization with no density assumption. The proof is based on the "generalized" Laplace trasform technique and refers to the classical Bessel second order differential equation. We also derived a matrix version of this generalized MY property.

References

(1) Bao, K.V., Noack, C., Characterizations of the generalized inverse Gaussian, asymmetric Laplace, and shifted (truncated) exponential laws via independence properties. arXiv 2107.01394 (2021), 1-12.

(2) Croydon, D.A., Sasada, M., Detailed balance and invariant measures for systems of locally-defined dynamics. arXiv 2007.06203 (2020), 1-48.

(3) Letac, G., Wesolowski, J., An independence property for the GIG and gamma laws. Ann. Probab. 28(3) (2000), 1371-1383.

(4) Letac, G., Wesolowski, J., About an extension of the Matsumoto-Yor property. (2022) - unpublished manuscript.

(5) Matsumoto, H., Yor, M., An analogue of Pitman's 2M-X theorem for exponential Wiener functionals. Part II: the role of the generalized inverse Gaussian laws Nagoya Math. J. 162 (2001), 65{86.

(6) Matsumoto, H., Yor, M., Interpretation via Brownian motion of some independence properties between GIG and gamma variables. Statist. Probab. Lett. 61 (2003), 253-259.