DISTRIBUTION OF A RANDOM FUNCTIONAL OF A FERGUSON-DIRICHLET PROCESS OVER THE UNIT SPHERE

Jiang, Dickey, and Kuo [ 12 ] gave the multivariate c -characteristic function and showed that it has properties similar to those of the multivariate Fourier transformation. We ﬁrst give the multivariate c -characteristic function of a random functional of a Ferguson-Dirichlet process over the unit sphere. We then ﬁnd out its probability density function using properties of the multivariate c -characteristic function. This new result would generalize that given by [ 11 ] .


Introduction
Ferguson [5] introduced the Ferguson-Dirichlet process and studied its applications to nonparametric Bayesian inference. He also showed that when the prior distribution is a Ferguson-Dirichlet process with parameter µ, then the posterior distribution, given the sample s 1 , s 2 , . . . , s n , is also a Ferguson-Dirichlet process having parameter µ + n j=1 δ s j , where δ s j denotes point mass at s j . The most natural use of random functionals of a Ferguson-Dirichlet process is to make Bayesian inferences concerning the parameters of a statistical population. Hence, the expression for the probability density function of any random functional of a Ferguson-Dirichlet process can be employed both for prior and posterior Bayesian analyses. Further applications related to the random functional can be seen in [3] and other references. For example, random means and random variances of a Ferguson-Dirichlet process can be used for smooth Bayesian nonparametric density estimation (see [15]) and for quality control problems (see [4] for further discussions), respectively. Research on the distribution of a random functional of a Ferguson-Dirichlet process has been ongoing for decades. A partial list of papers in this area are [2,3,8,9,11,12,14,16,17]. In particular, [11] gave the distribution of a random functional of a Ferguson-Dirichlet process over the unit circle. In this paper, we shall use the multivariate c-characteristic function, a tool given by [12], to generalize the result to the case over the unit sphere in three-dimension. In Section 2, we first review the definition of the multivariate c-characteristic function and some of its properties. We then compute a multivariate c-characteristic function of an interesting distribution. The multivariate c-characteristic function of the random mean of a Ferguson-Dirichlet process over the unit sphere is given in Section 3. Using the uniqueness property of the multivariate c-characteristic function, we then determine the distribution of the random mean of a Ferguson-Dirichlet process over the unit sphere. Conclusions are given in Section 4.

Multivariate c-characteristic function
Jiang [10] first gave a univariate c-characteristic function. Jiang, Dickey, and Kuo [12] generalized it to a multivariate c-characteristic function, which can be very useful when a distribution is difficult to deal with by traditional characteristic function. See [12] for detailed results. First, we state the definition of the multivariate c-characteristic function.
, and t · u is the inner product of t and u.
The above assumptions that c is positive and u has a bounded support are needed in [12, Lemma 2.2], which shows that, for any positive c, there is a one-to-one correspondence between g(t ; u, c) and the distribution of u. Next, we give the multivariate c-characteristic function of an interesting distribution in the next lemma.

Distribution of a random functional of a Ferguson-Dirichlet process over the unit sphere
First, we give a trivariate c-characteristic function expression of any trivariate random functional of a Ferguson-Dirichlet process over a Borel set Y in Euclidean space in the next lemma. µ on (Y, A). Then the trivariate c-characteristic function of w can be expressed as

bounded measurable function defined on a Borel set Y in Euclidean space n , and U is a Ferguson-Dirichlet process with parameter
Proof. For any k ≥ 2, let {B k1 , B k2 , . . . , B kk } be a partition of Y , b k j ∈ B k j , v k = max{volume(B k j ) | 1 ≤ j ≤ k}, and lim k→∞ v k = 0. Then (U(B k1 ), . . . , U(B kk )) follows a Dirichlet distribution with parameter (µ(B k1 ), . . . , µ(B kk )). In addition, and is 0, otherwise. Then lim k→∞ g k (x ) = h(x ) for all x ∈ Y , and w k = k j=1 g k (b k j )U(B k j ). The trivariate c-characteristic function of w k can be expressed as where is a Carlson's multiple hypergeometric function ( [1]), and the last equality can be obtained by [1, formula 6.6.5]. Therefore, the limit of the trivariate c-characteristic function of w k 's, as k approaches ∞, is In addition, by the Dominated Convergence Theorem, we have lim k→∞ w k = w . By [12, Theorem 2.4], we conclude that In the rest of this section, we study the random functional u = X x d U(x ), where X is the unit sphere in 3 . We use Lemma 3 in the following theorem to first establish the trivariate c-characteristic function of u.
, and U be a Ferguson-Dirichlet process over X with uniform measure µ as its parameter, where µ(X ) = c. Then the trivariate c-characteristic function of the random functional u = X x d U(x ) can be expressed as Proof. First, we give the following two equations, which are about Appell's notations and can be shown easily. Γ(a + n) = Γ(a)(a, n), (a, 2n) = 2 2n a 2 , n a + 1 2 , n .
By Lemma 3, we have The fifth identity can be obtained by Eqs. (2) and (3). The last identity follows from Eqs. (9) and (10).

Conclusions
In this paper, we obtain the trivariate c-characteristic function expression for a random functional of a Ferguson-Dirichlet process over any finite three-dimensional space. We also obtain the probability density function of the random functional of a Ferguson-Dirichlet process with uniform probability measure parameter over the unit sphere. This generalizes [11,Theorem 2].