In this paper we will deal with the determination of the inverse of a dichotomous probability distribution. In particular it will be shown that a dichotomous distribution admit inverse if and only if it corresponds to a random variable assuming values $(0,a)$, $a\in {ℝ}^{+}$. Moreover we will provide two general results about the behaviour of the inverse distribution relative to the power and to a linear transformation of a measure.

We study deformations of the free convolution arising via invertible transformations of probability measures on the real line T:μ ↦ Tμ. We define new associative convolutions of measures by $\mu {⊞}_T\nu =T^{-1}\left(T\mu ⊞T\nu \right)$. We discuss infinite divisibility with respect to these convolutions, and we establish a Lévy-Khintchine formula. We conclude the paper by proving that for any such deformation of free probability all probability measures μ have the Nica-Speicher property, that is, one can find their convolution power ${\mu }^{{⊞}_{T}s}$ for...

We study deformations of the classical convolution. For every invertible transformation T:μ ↦ Tμ, we are able to define a new associative convolution of measures by $\mu {*}_T\nu =T^{-1}(T\mu *T\nu )$. We deal with the $V_a$-deformation of the classical convolution. We prove the analogue of the classical Lévy-Khintchine formula. We also show the central limit measure, which turns out to be the standard Gaussian measure. Moreover, we calculate the Poisson measure in the $V_a$-deformed classical convolution and give the orthogonal polynomials...