An extension of Knight's theorem on Cauchy distribution (Q1102617)

Let X be a random variable with Cauchy distribution. The author describes a class of functions f(x), \(x\in R\), possessing the following property: The random variables X and f(X) are of the same type, i.e. \(P(f(X)<x)=P(\alpha +pX<x)\) for every \(x\in R\) and some \(a\in R\), \(p>0.\) A generalization of a result of Knight is obtained: If the random variables \(\xi\) and \[ k\xi +\alpha -\sum^{m}_{j=0}p_ j/(\xi - \gamma_ j), \] where \(k\geq 0\), \(\alpha\in R\), \(P_ j>0\), \(\gamma_ 0<\gamma_ 1<...<\gamma_ m\), are of the same type and for every \(a\in R\) and \(p>0\) the distribution of \(a+pX\) has no atom, then \(\xi\) and X are of the same type.