臺灣博碩士論文加值系統

本篇論文主要是提出一個新的甕模型：
有一個甕，一開始裝有b個黑球和w個白球，每次隨機取出m
個球，假設取出k個黑球及m-k個白球，那麼除了放回原被取
出的m個球之外，再多放ck個黑球及c(m-k)個白球至甕中，其
中c是某固定的正整數。從每次抽球到把球放回甕中的這一連
串的動作稱為抽一次球。我們將連續地抽n次球。
事實上，當m=c=1，這個新模型就是古典的Pólya-Eggenberger甕模型。定義甕裡的黑球比例為Xn。仿照前人研究Pólya-Eggenberger甕模型的方法，我們證明Xn形成一個martingale，並且強收斂到一個隨機變數X。雖然我們並未能求得X的分配為何，但我們證明了X具有絕對連續的特性。

In this thesis, we propose a new urn model as following. A single urn contains b black balls and w white balls. For each time, we randomly draw m balls and note their colors, said k black balls and m-k white balls. Return the drawn balls with additional ck black balls and c(m-k) white balls. Repeat n times. When m=c=1, this is the classical Pólya-Eggenberger urn model. Let Xn be the fraction of black balls. To investigate the asymptotic properties of our new urn model, we first show that Xn forms a martingale and converges a.s. to a random variable X. The distribution of X is then shown to be absolutely continuous.

1 Introduction
2 Pólya-Eggenberger Urn Model and Its Generalizations and Modifications
2.1 The Pólya-Eggenberger Urn Model
2.2 A Tenable Pólya-Eggenberger Urn Model
2.3 Penmantle Urn Model
2.4 Hill, Lane and Sudderth Urn Model
2.5 Cannibal Model
2.6 Ivchenko and Ivanov Urn Model
2.7 A Generalized Pólya's Urn Design (GPUD)
2.8 The Waiting Time Random Variable with Quota
3 A New Urn Model and Some Computational Studies
3.1 A New Model
3.2 Computational Studies
4 Martingale Property and Absolute Continuity of the Limit
4.1 Martingale
4.2 Absolute Continuity
5 Appendix
5.1 Figures
5.2 Programs

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