Probability Measures on Product Spaces
Probability Measures on Product Spaces
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产品空间的概率测度
DOI：
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发表时间：
2020
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影响因子：
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通讯作者：
A. Klenke
中科院分区：
文献类型：
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作者：
A. Klenke
为了对随机时间演化进行建模，典型的方法是在积空间上构造概率测度。粗略地说，第一步是采取概率度量来模拟初始分布。第二步，在不同的概率空间上，对一个时间步后的分布进行建模。然后在每个后续步骤中，在进一步的概率空间中，对给定完整历史的下一个时间步的随机状态进行建模。在形式层面上，我们考虑概率空间和这些空间之间的马尔可夫核的乘积。最后，利用Ionescu-Tulcea定理证明了整个过程可以在单个无限积空间上实现。此外，Kolmogorov的扩展定理表明，即使时间集不是离散的，也可以进行类似的构造。
In order to model a random time evolution, the canonical procedure is to construct probability measures on product spaces. Roughly speaking, the first step is to take a probability measure that models the initial distribution. In the second step, on a different probability space, the distribution after one time step is modeled. Then in each subsequent step, on a further probability space, the random state in the next time step given the full history is modeled. On a formal level, we consider products of probability spaces and Markov kernels between such spaces. Finally, the Ionescu-Tulcea theorem shows that the whole procedure can be realized on a single infinite product space. Furthermore, Kolmogorov’s extension theorem shows that a similar construction can be performed even if the time set is not discrete.