{"title":"Shooting randomly against a line in Euclidean and non-Euclidean spaces","authors":"E. Orsingher, Bruno Toaldo","doi":"10.1080/17442508.2012.749260","DOIUrl":null,"url":null,"abstract":"In this paper we study a class of distributions related to the r.v. , for different distributions of . The problem is related to the hitting point of a randomly oriented ray and generalizes the Cauchy distribution in different directions. We show that the distribution of solves the Laplace equation of order , possesses even moments of order , and has bimodal structure when is uniform. We study also a number of distributional properties of functionals of , including those related to the arcsine law. Finally we study the same problem in the Poincaré half-plane and this leads to the hyperbolic distribution of which the main properties are explored. In particular we study the distribution of hyperbolic functions of , the law of sums of i.i.d. r.v.'s and the distribution of the area of random hyperbolic right triangles.","PeriodicalId":49269,"journal":{"name":"Stochastics-An International Journal of Probability and Stochastic Processes","volume":"15 1","pages":"16 - 45"},"PeriodicalIF":0.9000,"publicationDate":"2014-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Stochastics-An International Journal of Probability and Stochastic Processes","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1080/17442508.2012.749260","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we study a class of distributions related to the r.v. , for different distributions of . The problem is related to the hitting point of a randomly oriented ray and generalizes the Cauchy distribution in different directions. We show that the distribution of solves the Laplace equation of order , possesses even moments of order , and has bimodal structure when is uniform. We study also a number of distributional properties of functionals of , including those related to the arcsine law. Finally we study the same problem in the Poincaré half-plane and this leads to the hyperbolic distribution of which the main properties are explored. In particular we study the distribution of hyperbolic functions of , the law of sums of i.i.d. r.v.'s and the distribution of the area of random hyperbolic right triangles.
期刊介绍：
Stochastics: An International Journal of Probability and Stochastic Processes is a world-leading journal publishing research concerned with stochastic processes and their applications in the modelling, analysis and optimization of stochastic systems, i.e. processes characterized both by temporal or spatial evolution and by the presence of random effects.
Articles are published dealing with all aspects of stochastic systems analysis, characterization problems, stochastic modelling and identification, optimization, filtering and control and with related questions in the theory of stochastic processes. The journal also solicits papers dealing with significant applications of stochastic process theory to problems in engineering systems, the physical and life sciences, economics and other areas. Proposals for special issues in cutting-edge areas are welcome and should be directed to the Editor-in-Chief who will review accordingly.
In recent years there has been a growing interaction between current research in probability theory and problems in stochastic systems. The objective of Stochastics is to encourage this trend, promoting an awareness of the latest theoretical developments on the one hand and of mathematical problems arising in applications on the other.
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