%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 }