Which distributions have the Matsumoto-Yor property?

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.

the Kummer distribution of type 2 : The corresponding laws of U and V are where a, b > 0 and λ ∈ R.

The results
Theorem 2.1 Let (X, Y ) be a couple of independent positive r.v.'s with densities p X and resp. p Y . It is supposed that p X and p Y are positive and that log p X and log p Y are locally integrable over ]0, ∞[. Suppose that U and V defined by (2.1) are independent, then the densities of X and Y are given by (1.5), resp. (1.6). Moreover, U ∼ K (2) (a, b, c) and V ∼ γ(b, c).
Remark 2.2 1. Keeping the notation given in the Introduction, Theorem 2.1 means that D log (f ) = p X (x)dx ⊗ p Y (y)dy, a, b, c > 0 .
2. It is clear that (X, Y ) = log(1 + 1/V ′ ), − log U ′ with : It is easy to verify that log p X and log p Y are locally integrable on ]0, ∞[ if and only if log p U ′ and log p V ′ are locally integrable on ]0, 1[ and ]0, ∞[ respectively. Using usual calculations, the formulation of Matsumoto-Yor independence property related to U, V, U ′ and V ′ is the following : suppose that U and V are independent and U ′ and This gives a characterization of the Kummer distributions. Observe that we retrieve by the way the following convolution formula mentioned in [2] : be a couple of independent positive r.v.'s with densities p X and resp. p Y such that p X and p Y are positive, log p X and log p Y are locally integrable over are independent, then the densities of Y and X are given by (1.8) and resp. (1.9). Moreover (1.10) and (1.11) are the densities of U and resp. V .
for interpretations and details. We easily deduce from definitions : Note that log p X and log p Y are locally integrable on ]0, ∞[ if and only if log p X ′ and log p Y ′ are locally integrable on ]0, 1[. In this new setting Theorem 2.3 takes the following form.
a, b, λ > 0 and k is the normalizing constant.
In particular U * and V * are independent with densities the right-hand side of (1.5) and (1.6) respectively, with a = a 0 , b = b 0 and c = c 0 .

Proof of Theorem 2.1
Suppose that X and Y are independent, that the functions log p X and log p Y are locally integrable over ]0, ∞[, and that U and V are independent. Our approach is direct and is based on the calculation of the densities of (U, V ) and (X, Y ) using (2.1) and (2.2). This leads to two functional equations involving p X , p Y , p U and p V .
Then the following functional equations hold : Under the above assumptions, the density function of (U, V ) is and the one of (X, Y ) is : Replacing (z, w) by (w, z) in (3.10) leads to : First, we divide the left-hand side of (3.10) by the left-hand side of (3.11) and second, we take the logarithm, we obtain : . Therefore, .

Lemma 3.2
The functions H, α, F and β are of class C 1 .
The proof of Lemma 3.2 is postponed in a special subsection 3.2 devoted to this problem. Denote φ(s, t) := g(s) − g(s + t), g(t) − g(s + t) . Then (3.8) can be written as : . This leads us to consider functional equations of the type : and the goal is to give sufficient conditions so that θ and G are of class C 1 . Theorem 2.3 will be proved using the above approach. Consequently, V ∼ γ(M + 1, −2N).

Lemma 3.4 There exists
Proof of Lemma 3.4 Taking the u-derivative in (3.8) we get : Let us rewrite the terms in the right-hand side of (3.22), we have .
Recall that β = log p Y and p Y is a density function. Then integrating the previous identity gives directly (3.21).
Lemma 3.5 The density functions of U and X are respectively given by : (I, J ∈ R and N < 0) and this proves that U follows the Kummer distribution Proof of Lemma 3.5 a) It is clear that using the definition of g, (3.26) may be written as As a consequence,

Auxiliary results
In this section we give a theoretical setting which allows to prove that the pairs (H, α) and (F, β) (resp. (F δ , β)) introduced in Lemma 3.1 (resp. relation (3.39)) are of class C 1 .
2) It is clear that (3.35) and (3.34) imply : Consider any x 1 < x 0 < x 2 in V (x 0 ) and suppose that G is locally integrable. We can integrate (3.36) over (x 1 , x 2 ) with respect to x: After the change of variable s = ψ 1 (x, y) in the first integral and the change of variable t = ψ 2 (x, y) in the second integral, we get : Taking absolute values in (3.36) implies that all the previous integrals are finite. The case where θ is locally integrable can be handled similarly.
Since the left-hand side of (3.38) is continuous in y (because ψ is C 1 ), the function θ is continuous. From the continuity of θ and (3.34) we deduce that G is continuous (because φ is continuous). Consequently, the left-hand side of (3.38) is a C 1 function in y, hence θ is C 1 . We deduce, again from (3.34), that G is C 1 . Our approach has been inspired by the one of [5].
From now on, we consider the particular cases ξ = f and ξ = g. We have :

Proof of Theorem 2.3
Let us assume that X and Y are independent and U and V are independent. Since f δ is equal to its inverse then X = f δ (U + V ) and Y = f δ (U) − f δ (U + V ). Reasoning as in the proof of Lemma 3.1, we easily get : , F δ := log k and β := log p Y .
With ξ := f δ , we get : Consequently, we deduce from (3.39) and Lemma 3.7 that β and F δ are of class C 1 . Next, taking the u-derivative in (3.39) leads to : c) Combining (3.41) with (3.42) we deduce : Integrating the above relation and setting v = f δ (u) we obtain : where C > 0. Recall that β = log p Y , then Note that (X, Y ) = f δ (U + V ), f δ (U) − f δ (U + V ) and recall that X, Y are independent and U, V are independent. Applying (3.44) with Y instead of V gives : Then the computation of the densities of U and X is straightforward as in the proof of Theorem 2.1 and we have the desired result with A = b − 1, B ′ = −λ, B = −a, A ′ = b − 1.