The Minkowski length of a spherical random vector

The Rayleigh distribution represents the Euclidean length of a two-dimensional random vector with normally distributed components that are independent, while for the case of a three-dimensional random vector, its length distributed by the well-known Maxwell–Boltzmann distribution. In this letter, we generalize these results in two ways, into the world of elliptical distributions and for general L^p spaces. We present the distribution of the length of an n-dimensional random vector whose components are mutually dependent and symmetrically distributed in L^p spaces. The results show that such distribution has explicit form, which allows computing its moments. Similar to the Rayleigh distribution, the presented distribution can also be useful to model risks. Thus, we derive important risk measures for the investigated distribution.