The purpose of the present paper is to prove that (1) the ratio of two independent stable random variables with exponent d and skewness 0 is equal in distribution to a standard Cauchy multiplied by $sin(pi d R / 2) / sin( pi d (1-R)/2 )^1/d$, where R is a Uniform0,1 random variabel; and (2) the same ratio except with skewness parameter beta is equal to $W_1(\beta) W_2(\beta) \sin(pi d R / 2) / \sin( pi d (1-R)/2 )^1/d$, where $W(0)$ is a standard Cauchy, and $W(\beta) = W(0) \cos(\beta/2) + \sin( \beta/2 )$.
%0 Book Section
%1 shcolnick1985ratio
%A Shcolnick, S. M.
%B Stability Problems for Stochastic Models: Proceedings of the 8th International Seminar held in Uzhgorod, USSR, Sept. 23--29, 1984
%C Berlin, Heidelberg
%D 1985
%E Kalashnikov, Vladimir V.
%E Zolotarev, Vladimir M.
%I Springer Berlin Heidelberg
%K Cauchy_distribution probability_theory ratio_of_random_variables stable_distributions
%P 349--354
%R 10.1007/BFb0074827
%T On the ratio of independent stable random variables
%U http://dx.doi.org/10.1007/BFb0074827
%X The purpose of the present paper is to prove that (1) the ratio of two independent stable random variables with exponent d and skewness 0 is equal in distribution to a standard Cauchy multiplied by $sin(pi d R / 2) / sin( pi d (1-R)/2 )^1/d$, where R is a Uniform0,1 random variabel; and (2) the same ratio except with skewness parameter beta is equal to $W_1(\beta) W_2(\beta) \sin(pi d R / 2) / \sin( pi d (1-R)/2 )^1/d$, where $W(0)$ is a standard Cauchy, and $W(\beta) = W(0) \cos(\beta/2) + \sin( \beta/2 )$.
%@ 978-3-540-39686-4
@inbook{shcolnick1985ratio,
abstract = {The purpose of the present paper is to prove that (1) the ratio of two independent stable random variables with exponent d and skewness 0 is equal in distribution to a standard Cauchy multiplied by $sin(pi d R / 2) / sin( pi d (1-R)/2 )^{1/d}$, where R is a Uniform[0,1] random variabel; and (2) the same ratio except with skewness parameter beta is equal to $W_1(\beta) W_2(\beta) \sin(pi d R / 2) / \sin( pi d (1-R)/2 )^{1/d}$, where $W(0)$ is a standard Cauchy, and $W(\beta) = W(0) \cos(\pi \beta/2) + \sin( \pi \beta/2 )$.},
added-at = {2016-07-01T08:15:36.000+0200},
address = {Berlin, Heidelberg },
author = {Shcolnick, S. M.},
biburl = {https://www.bibsonomy.org/bibtex/2c6ca27e39c932c856b18f0fc6dcca272/peter.ralph},
booktitle = {Stability Problems for Stochastic Models: Proceedings of the 8th International Seminar held in Uzhgorod, USSR, Sept. 23--29, 1984},
doi = {10.1007/BFb0074827},
editor = {Kalashnikov, Vladimir V. and Zolotarev, Vladimir M.},
interhash = {59377bbce1fc177b60c495a03abf4870},
intrahash = {c6ca27e39c932c856b18f0fc6dcca272},
isbn = {978-3-540-39686-4},
keywords = {Cauchy_distribution probability_theory ratio_of_random_variables stable_distributions},
pages = {349--354},
publisher = {Springer Berlin Heidelberg},
timestamp = {2016-07-01T09:16:25.000+0200},
title = {On the ratio of independent stable random variables},
url = {http://dx.doi.org/10.1007/BFb0074827},
year = 1985
}