2012
DOI: 10.3150/10-bej325
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Independence properties of the Matsumoto–Yor type

Abstract: We define Letac-Wesolowski-Matsumoto-Yor (LWMY) functions as decreasing functions from (0, ∞) onto (0, ∞) with the following property: there exist independent, positive random variables X and Y such that the variables f (X + Y ) and f (X) − f (X + Y ) are independent. We prove that, under additional assumptions, there are essentially four such functions. The first one is f (x) = 1/x. In this case, referred to in the literature as the Matsumoto-Yor property, the law of X is generalized inverse Gaussian while Y … Show more

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Cited by 22 publications
(52 citation statements)
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“…The generalized inverse Gaussian (hereafter GIG) distribution with parameters p ∈ R, a > 0, b > 0 has density g p,a,b (x) = (a/b) p/2 2K p ( √ ab) x p−1 e − 1 2 (ax+b/x) , x > 0, where K p is the modified Bessel function of the third kind. For details on GIG and Kummer distributions see [4,5,11] and references therein, where one can see for instance that these distributions are involved in some characterization problems related to the so-called Matsumoto-Yor property.…”
Section: Introductionmentioning
confidence: 99%
“…The generalized inverse Gaussian (hereafter GIG) distribution with parameters p ∈ R, a > 0, b > 0 has density g p,a,b (x) = (a/b) p/2 2K p ( √ ab) x p−1 e − 1 2 (ax+b/x) , x > 0, where K p is the modified Bessel function of the third kind. For details on GIG and Kummer distributions see [4,5,11] and references therein, where one can see for instance that these distributions are involved in some characterization problems related to the so-called Matsumoto-Yor property.…”
Section: Introductionmentioning
confidence: 99%
“…There is a very interesting family of ψ's which was introduced in [14]. From the probabilistic point of view, this family is somehow related to the so-called Matsumoto-Yor property.…”
Section: Introductionmentioning
confidence: 99%
“…It is has been proved in Letac and Wesołowski (2000) and Matsumoto and Yor (2001) that the function f 0 (x) = 1/x is a LWMY function and the related distributions are Generalized Inverse Gaussian and Gamma. Under additional assumptions, Koudou and Vallois (2012) have shown that there exist 4 classes of LWMY functions including the one generated by f 0 . The three other classes are generated respectively by f 1 (x) = 1 e x − 1 , g 1 (x) = f −1 1 (x) and f * δ (x) = ln ( e x +δ−1 e x −1 ) where x > 0 and δ > 0.…”
mentioning
confidence: 98%
“…3) Concerning the last class of LMWY functions which is generated by the function f * δ , with a suitable change of variable Koudou and Vallois (2012) have shown that the MatsumotoYor independence property takes the following form: we look for two independent random variables X and Y such that f δ (XY ) and f δ (X) f δ (XY ) are independent, where:…”
mentioning
confidence: 99%
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