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.
wherek⩾0,p j >0,α∈ℝ, γ0<⋯<γ m .
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References
Dunau, J. L., and Senateur, H. (1987). An elementary proof of the Knight-Meyer characterization of the Cauchy distribution. To appear inJ. Multivar. Anal.
Kemperman, J. H. B. (1975). The ergodic behaviour of a class of real transformations. InStochastic Processes and Related Topics.I (Proc. Summer Research Ins. on Statistical Inference for Stochastic Processes, Indiana University, Bloomington, 1974), Academic, New York, pp. 249–258.
Knight, F. B. (1976). A characterisation of the Cauchy type.Proc. Am. Math. Soc., 130–135.
Knight F. B., and Meyer, P. A. (1976). Une caractérisation de la loi de Cauchy.Z. Warscheinlichkeitstheorie verw. Geb. 34, 129–134.
Letac, G. (1977). Which functions preserve Cauchy laws?Proc. Am. Math. Soc. 67, 277–286.
Williams, E. J. (1969). Cauchy-distributed functions and a characterization of the Cauchy distribution.Ann. Math. Stat.,40, 1083–1085.
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Hassenforder, C. An extension of Knight's theorem on Cauchy distribution. J Theor Probab 1, 205–209 (1988). https://doi.org/10.1007/BF01046935
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DOI: https://doi.org/10.1007/BF01046935