Abstract
If μ is a probability on ℝ, the set of distributions ofa+pX wherea∈ℝ,p>0 andX has distribution μ is called the type of μ. F. B. Knight has shown that if a type has no atom and if it is invariant byi:x↦−1/x, the type must be the Cauchy one. We show here thati can be replaced by any Cayley nonaffine function. $$\varphi (x) = kx + \alpha - \sum\limits_{j = 0}^m {p_j (x - \gamma _j )^{ - 1} } $$ wherek⩾0,pj>0,α∈ℝ, γ0<⋯<γm.
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