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 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
 [1] Bagchi, A. and Pal, A. K. (1985). Asymptoic Normality in the Generalized Pólya-Eggenberger Urn Model, with an Application to Computer Data Structures. SIAM Journal of Algebraic Discrete Methods. 6. 394-405.[2] Billingsley, P. (1995). Probability and Measure. Third Edition. New York: John Wiley and Sons.[3] Blom, G. and Holst, L. (1991). Embedding Procedures for Discrete Problems in Probability. The Mathematical Scientist. 16. 29-40.[4] Chung, K. L. (1974). A Course in Probability Theory. Second Edition. San Diego: Academic Press.[5] Eggenberger, F. and Pólya, G. (1923). Über die Statistik Verketetter Vorgänge. Zeitschrift für Angewandte Mathematik und Mechanik. 1. 279-289.[6] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Second Edition. New York: John Wiley and Sons. II.[7] Gouet, R. (1989). A Martingale Approach to Strong Convergence in a Generalized Pólya-Eggenberger Urn Model. Statistics and Probabilitty Letters. 8. 225-228.[8] Gouet, R. (1993). Martingale Functional Central Limit Theorems for a Generalized Pólya Urn. The Annals of Probability. 21. 1624-1639.[9] Green, R. F. (1980). The Cannibal's Urn. Unpublished Manuscript.[10] Hall, P. and Heyde, C. C. (1980). Martingale Limit Theory and Its Application. New York: Academic Press.[11] Hill, B., Lane, D. and Sudderth, W. (1980). A Strong Law for Some Generalized Urn Processes. The Annals of Probability. 8. 214-226.[12] Ivchenko, G. I. and Ivanov, V. A.(1995). Decomposable Statistics in Inverse Urn Problems. Discrete Mathematics and Applications. 5. 159-172.[13] Johnson, N. L. and Kotz, S. (1977). Urn Models and Their Application. New York: John Wiley and Sons. 176-181.[14] Kotz, S. and Balakrishnan, N. (1997). Advances in Urn Models During the Past Two Decades. Advances in Combinatorial Methods and Applications to Probability and Statistics. 203-257.[15] Ling, K. D. (1993). Sooner and Later Waiting Yime Distributions for Frequency Quota Defined on a Pólya-Eggenberger Urn Model. Soochow Journal of Mathematics. 19. 139-151.[16] Maistrov, L. E. (1974). Probability Theory: a Historical Sketch. New York and London: Academic Press.[17] Medvedev, Yu. I. (1970). Some Theorems on Asymptotic Distribution of the Chi-squared Statistic. Soviet Mathematical Doklady. 192. 987-989.[18] Pemantle, R. (1990). A Time-Dependent Version of Pólya's Urn. Journal of Theoretical Probability. 3. 627-637.[19] Pittel, B. (1987). An Urn Model for Cannibal Behavior. Journal of Applied Probability. 24. 522-526.[20] Stewart, I. (1989). Galois Theory. Second Edition. New York: Chapman and Hall. %21-22[21] Wei, C. Z. (1993). Martingale Transforms with Non-atomic Limits and Stochastic Approximation. Probability Theory and Related Fields. 95. 103-114.[22] Wei, L. J. (1979). The Generalized Pólya's Urn Design for Sequential Medical Trials, Annals of Statistics. 7. 291-296.[23] Wheeden, R. L. and Zygmund, A. (1977). Measure and Integral:An Introduction to Real Analysis. New York: Dekker, M.
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