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We propose a bivariate model for a pair of dependent unit vectors which isgenerated by Brownian motion. Both marginals have uniform distributions on thesphere, while the conditionals follow so-called ``exit'' distributions. Someproperties of the proposed model, including parameter estimation and a pivotalstatistic, are investigated. Further study is undertaken for the bivariatecircular case by transforming variables and parameters into the form of complexnumbers. Some desirable properties, such as a multiplicative property andinfinite divisibility, hold for this submodel. Two estimators for the parameterof the submodel are studied and a simulation study is carried out toinvestigate the finite sample performance of the estimators. In an attempt toproduce more flexible models, some methods to generalize the proposed model arediscussed. One of the generalized models is applied to wind direction data.Finally, we show how it is possible to construct distributions in the plane andon the cylinder by applying bilinear fractional transformations to the proposedbivariate circular model.

A distribution for a pair of unit vectors generated by Brownian motion