波利亚随机游走定理证伪
首发时间:2020-03-19
摘要:随机游走是概率论与随机过程学科中用于描述动态随机现象的一种基本随机过程。液体中悬浮微粒的布朗运动、光纤陀螺中的随机漂移误差和股票市场中的价格波动等动态随机现象均可用随机游走模型进行描述。本文从波利亚随机游走定理出发,直接推出了波利亚随机游走定理与随机游走统计特性和中心极限定理相悖的结论,从而用归谬法证明了波利亚随机游走定理的命题为假。本文依据随机游走样本轨道算数平均值和大数定律,推导出了揭示随机游走现象、特征及规律的位移公式,得出了一维随机游走位移与时间成正比的结论。
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Disproving Polya\'s Random Walk Theorem
Abstract:Random walk is a basic random process used to describe dynamic random phenomena in probability theory and stochastic process. Dynamic random phenomena such as Brownian motion of suspended particles in liquids, random walk errors in fiber optic gyroscopes, and price fluctuations in the stock market can be described using random walk models. This paper applies the Induction to Absurdity from Polya \'s random walk theorem to derive the conclusion that Polya \'s random walk theorem contradicts the statistical characteristics of the random walk and the central limit theorem, thereby proving that the proposition of Polya \'s random walk theorem is false. Based on the Law of Large Numbers and the arithmetic mean of the sample path of random walk , this paper derives a displacement formula that reveals the phenomenon, characteristics and rules of random walk, and draws the conclusion that the one-dimensional random walk displacement is proportional to time.
Keywords: Random Walk Brownian motion Recurrence
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