A new multivariate transform and the distribution of a random functional of a Ferguson–Dirichlet process

Distribution of functionals of a Ferguson–Dirichlet process over ann-dimensional ball

“…We then provide the exact probability density functions of these random means. These results generalize those given by Jiang [10], Jiang et al [11], and Jiang and Kuo [12] over two or three dimensional distributions. Lastly, we further extend the results to the random mean of a spherically symmetric Ferguson-Dirichlet process with parameter measure over an n-dimensional ellipsoidal solid and that over an n-dimensional ellipsoidal surface.…”

“…Since g(t; u, c) is a function of |t| and c, we have that g(T ′ t; u, c) is a function of |T ′ t| = |t| and c. That is, T u and u have the same c-characteristic function. By Lemma 2.2 of Jiang et al [11], T u and u have the same distribution for any orthogonal matrix T . Therefore, u has a spherically symmetric distribution by Definition 2.2.…”

“…Then, by Theorem 4.1 of Jiang et al [11], we have g(t; T u, c) = g(T ′ t; u, c) where T ′ is the transpose of T . Since g(t; u, c) is a function of |t| and c, we have that g(T ′ t; u, c) is a function of |T ′ t| = |t| and c. That is, T u and u have the same c-characteristic function.…”

“…Some important properties, e.g., uniqueness and convergence theorems, of the c-characteristic function can be seen in [11]. In the following, we give definitions of a spherically symmetric distribution and an ellipsoidally symmetric distribution.…”

“…In Section 2, we give a new property of the c-characteristic function, which can solve many problems that are difficult to manage using the traditional characteristic function (see [9][10][11]), for a spherically symmetric distribution. This property can be used to easily determine whether a random vector (variable) has a spherically symmetric distribution.…”