Abstract
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 Wesolowski (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.
Highlights
Consider a decreasing and bijective function ξ : ]0, ∞[→]0, ∞[
If X, Y are non-Dirac, positive and independent random variables with law μX and μY respectively, we say that the triplet (ξ, μX, μY ) has the Matsumoto-Yor property if the r.v.’s : U := ξ(X + Y ), V := ξ(X ) − ξ(X + Y )
Define log (ξ) (resp. 2(ξ)) as the subset of (ξ) so that X and Y have densities whose logarithms are locally integrable over ]0, ∞[
Summary
Consider a decreasing and bijective function ξ : ]0, ∞[→]0, ∞[. If X , Y are non-Dirac, positive and independent random variables with law μX and μY respectively, we say that the triplet (ξ, μX , μY ) has the Matsumoto-Yor property if the r.v.’s :. Let (ξ) denote the set of all possible laws of (X , Y ) such that (ξ, μX , μY ) has the Matsumoto-Yor property. For any x > 0, define y = h(α, β, δ)(x) as the unique y > 0 so that eα(x)eβ ( y) = δ Under additional assumptions, it has been proved in [1] that. For a = b it has been proved in [4] that if X and Y are two independent r.v.’s, X ∼ GIG(−p, a, b) and. Where a, b, c > 0 and C∗ is the normalizing constant (in the sequel, C∗ stands for the unique positive constant so that the related function is a density).
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