多變量d轉換的一些應用

Abstract

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.

書名頁\r\n謝辭\r\nContents\r\nAbstract i\r\n中文摘要 ii\r\n1. Introduction 1\r\n1.1 The c-characteristic function 1\r\n1.2 The Ferguson-Dirichlet process 2\r\n1.3 Compatible conditional distributions 4\r\n1.4 Inverse Bayes formula 6\r\n2. Notations and useful equations 7\r\n2.1 Notations 7\r\n2.2 Useful equations 8\r\n3. The c-characteristic function 12\r\n3.1 The univariate c-characteristic function 12\r\n3.2 The multivariate c-characteristic function 13\r\n3.3 Inversion formulas of a univariate c-characteristic function 16\r\n3.4 Density construction through Fourier transformation 24\r\n3.5 Distributions of linear combinations of the components of a Dirichlet random vector 25\r\n4. The Ferguson-Dirichlet process 31\r\n4.1 Random functionals of a Ferguson-Dirichlet process 31\r\n4.2 The Ferguson-Dirichlet process over n-dimensional sphere 41\r\n4.3 Proof of Theorem 4.5 48\r\n4.4 Proof of Theorem 4.6 55\r\n5. Compatible conditional distributions 59\r\n5.1 The bivariate discrete case 59\r\n5.2 The trivariate discrete case 77\r\n5.3 The general discrete case 91\r\n5.4 The bivariate continuous case 93\r\n5.5 The general continuous case 98\r\n6. Generalized inverse Bayes formula 102\r\n6.1 The finite and discrete case 102\r\n6.2 The continuous case 108\r\n6.3 Algorithm 115\r\n7. Conclusions 119\r\nReferences 121