Jiang (1997) 首先提出多變量d轉換與其性質。利用多變量d轉換，我們可以定義新式的特徵函數，並且稱它們是多變量d特徵函數。在這篇論文中，我們將使用多變量d特徵函數來證明在普通的條件下，Dirichlet隨機向量的線性組合會分配收斂（converge in distribution）到一個對稱的分配。此外，當給定一個分配函數的多變量d特徵函數，我們將建構一個方法來決定此分配函數。另一方面，我們將證明多變量d特徵函數擁有很多類似傳統的特徵函數的性質。

　　A multivariate d-transformation and its properties were first given by Jiang (1997). By means of the multivariate d-transformations, we can define new kinds of characteristic functions and call them multivariate d-characteristic functions. In this thesis, we will use the multivariate d-characteristic function to show that the linear combinations of Dirichlet random vectors, under regularity conditions, converge in distribution to a spherical distribution. Moreover, We will construct a method for constructing the distribution function with a given multivariate d-characteristic function. In addition, we will show that the multivariate d-characteristic function has many properties which are similar to those of the traditional characteristic function.

Apostol, T. M. (1974), Mathematical Analysis, 2nd ed., Addison-Wesley.
Carlson, B. C. (1977), Special Functions of Applied Mathematics, Academic Press, New York.
Chung, Kai Lai (1968), A Course in Probability Theory, Harcourt,Brace and World, New York.
Dickey, J. M. and Jiang, T. J. (1998), "Filtered-Variate Prior Distributions for Histogram Smoothing," Journal of the American Statistical Association, 93, pp. 651--662.
Erdelyi, A. (1953) (Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F. G.), Higher Transcendental Functions, vol. I, McGraw-Hill, New York.
Friedberg, S. H., Insel, A. J., and Spence, L. E. (1992), Linear Algebra, 2nd ed., Prentice-Hall, New York.
Gradshteyn, I. S. and Ryzbik, I. M.; Alan Jeffrey, editor; Translated from the Russian by Scripta Technica, Inc. (2000), Table of Integrals, Series, and Products, 6th ed., Academic Press, Boston.
Hogg, Robert V. and Craig, Allen T. (1995), Introduction to Mathematical Statistics, 5th ed., Prentice-Hall, Englewood Cliffs, New Jersey.
Jiang, J. (1988), "Starlike functions and linear functions of a Dirichlet distributed vector," SIAM J. Math. Anal., 19, pp. 390--397.
Jiang, T. (1991), "Distribution of random functional of a Dirichlet process on the unit disk," Statistics and Probability Letters, 12, pp. 263--265.
Jiang, J. (1997), "Multivariate d-transformations with applications," Taiwan National Science Council final report (NSC 85--2121--M--004--007).
Jiang, T. (2002), "A new multivariate transformation and distribution of random functional of a Dirichlet process on the solid bounded by an ellipse." To be published.
Lord, R. D. (1954), "The use of the Hankel transformations in statistics. I. General theory and examples," Biometrika, 41, pp. 44--55.
Marsden, Jerrold E. and Hoffman, Michael J. (1993),Elementary Classical Analysis, 2nd ed., W. H. Freeman and Company, New York.
Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I. (1986), Integrals and Series, vol. 3, Gordon and Breach Science Publishers, New York.
Roussas, George G. (1997), A Course in Mathematical Statistics, 2nd ed., Academic Press, San Diego, Calif.