This paper is concerned with random vectors \(\mathbf X\) possessing the stochastic representation \({\mathbf X}=R{\mathbf U}\) where \(R\) is a positive random radius and \({\mathbf U}\) is an \(L_p\)-norm generalised symmetrised Dirichlet (LpGSD) random vector independent of \(R\). The authors provide some basic distributional properties of the vectors \(\mathbf X\) and study dependence and asymptotic dependence of LpGSD distributions. Further, conditional limiting theorems motivated by some result of S.M. Berman are derived. It is surprising that the standard Kotz Type I LpGSD distribution approximates a large subclass of the conditional distributions of LpGSD random vectors. In addition, in the case when the random radius \(R\) is regularly varying, the asymptotic tail behaviour is discussed.