Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability
Convergence of random series and the rate of convergence of the strong law of large numbers in game-theoretic probability
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博弈论概率中随机级数的收敛性和大数强定律的收敛率
DOI：
10.1016/j.spa.2011.10.011
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发表时间：
2012
影响因子：
1.4
通讯作者：
A.
中科院分区：
文献类型：
--
作者：
Miyabe;K. and Takemura;A.
在Shafer和Vovk（2001）[24]的博弈论概率框架下，我们给出了随机级数的收敛性和强大数定律的收敛速度的统一处理.我们认为游戏的二次对冲以及更一般的弱对冲。后者对应于存在一个绝对矩的顺序小于2的测量理论框架。证明了中心随机序列的收敛性与套期保值价格序列的收敛性之间的精确关系。当解释的测度理论框架，这些结果的特点是收敛的条件绝对矩系列的鞅。为了证明这些结果，我们得出一些基本结果的现实，谁是一个球员在协议的博弈论概率的确定性策略。这是特别感兴趣的，因为现实的战略没有任何对应的测量理论框架，但他们可以用来证明结果，可以解释的测量理论框架。
We give a unified treatment of the convergence of random series and the rate of convergence of the strong law of large numbers in the framework of game-theoretic probability of Shafer and Vovk (2001) [24]. We consider games with the quadratic hedge as well as more general weaker hedges. The latter corresponds to the existence of an absolute moment of order smaller than 2 in the measure-theoretic framework. We prove some precise relations between the convergence of centered random series and the convergence of the series of prices of the hedges. When interpreted in the measure-theoretic framework, these results characterize the convergence of a martingale in terms of the convergence of the series of conditional absolute moments. In order to prove these results we derive some fundamental results on deterministic strategies of Reality, who is a player in a protocol of game-theoretic probability. It is of particular interest, since Reality’s strategies do not have any counterparts in the measure-theoretic framework, ant yet they can be used to prove results which can be interpreted in the measure-theoretic framework.