傳統常數補充 (constant replacement) 的波利亞甕模型具備已知的漸近分佈，然而帶有隨機補充機制 (randomized replacement) 的廣義模型卻難以求得極限的封閉形式解。本文將離散時間的雙色隨機甕模型嵌入至連續時間的馬可夫分支過程 (Markov branching processes) 中，並利用 Doob 平賭收斂定理嚴格證明了白球比例的幾乎確實收斂。同時，我們推導出極限分佈之拉普拉斯轉換，指出了由隨機機制所導致的解析解瓶頸。
為此，本文選擇採用蒙地卡羅模擬與 Kolmogorov-Smirnov (K-S) 檢定進行實證分析。結果顯示，過去文獻提出的理論混合模型無法精確捕捉真實極限；相反地，我們提出由實驗數據直接估計參數的「資料驅動之 Beta 分佈 (data-driven Beta distribution)」展現了較佳的統計配適度。此發現為隨機甕模型複雜的極限分佈，提供了一個穩健且具實用價值的近似解。

While classical Pólya urn models with constant replacement have wellestablished asymptotic distributions, generalized models with randomized replacement mechanisms lack tractable closed-form limits. This thesis investigates a twocolor randomized Pólya urn model by embedding its discrete-time dynamics into continuous-time Markov branching processes. Using Doob’s Martingale Convergence Theorem, we rigorously prove the almost sure convergence of the urn proportion and derive the Laplace transform of the limit, explicitly identifying the analytical bottleneck caused by stochastic reinforcement.
To this end, this thesis opts to employ Monte Carlo simulations and KolmogorovSmirnov (K-S) testing for empirical analysis. Our results show that while theoretically proposed mixture models fail to adequately capture the true limit, a data-driven Beta distribution—with shape parameters estimated directly from experimental data—demonstrates a better statistical fit. This approach offers a robust and practical approximation for the complex limiting distribution of randomized urn processes.

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