查普曼-科尔莫戈罗夫等式

在数学之概率论中，尤其是随机过程理论中，查普曼-科尔莫戈罗夫等式是一个重要的结论。它将一个随机过程的几个不同维的联合分布函数联系在一起。

假设 { f_i } 是一个随机过程，即一个随机变量集合（每个元素对应一个只命名不排序的索引）。记

p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})

为从f₁到f_n的各随机变量的联合分布函数，则查普曼-科尔莫戈罗夫等式为：

p_{i_{1},\ldots ,i_{n-1}}(f_{1},\ldots ,f_{n-1})=\int _{-\infty }^{\infty }p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})\,df_{n}

也就是说，这是一个直接定义在干扰随机变量上的条件概率。（注意这里各随机变量的顺序不重要）

该公式名称来自数学家西德尼·查普曼和安德雷·柯尔莫哥洛夫。

特化为马尔可夫链

如果随机过程特定为马尔可夫链，查普曼-科尔莫戈罗夫等式就是关于转移概率的公式。在马尔可夫链中，随机变量在一个按时间排序的数组 $i_{1}<\ldots <i_{n}$ 中。按马尔可夫性质（无记忆性质）,

p_{i_{1},\ldots ,i_{n}}(f_{1},\ldots ,f_{n})=p_{i_{1}}(f_{1})p_{i_{2};i_{1}}(f_{2}\mid f_{1})\cdots p_{i_{n};i_{n-1}}(f_{n}{\mid }f_{n-1}),

（其中条件概率 $p_{i;j}(f_{i}{\mid }f_{j})$ 是 $i>j$ 时间的转移概率。查普曼-科尔莫戈罗夫等式简化为：

p_{i_{3};i_{1}}(f_{3}{\mid }f_{1})=\int _{-\infty }^{\infty }p_{i_{3};i_{2}}(f_{3}\mid f_{2})p_{i_{2};i_{1}}(f_{2}\mid f_{1})df_{2}.

如果马尔可夫链的状态空间的概率分布是离散的，查普曼-科尔莫戈罗夫等式可表示为（可到无穷维的）矩阵相乘：

P(t+s)=P(t)P(s)\,

（其中 $P(t)$ 是转移矩阵， $X_{t}$ 是t时间的系统状态），则对于系统状态空间中的任意两个点i和j：

P_{ij}(t)=P(X_{t}=j\mid X_{0}=i).

参考文献

The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography. Kolmogorov School. Ph.D. students and descendants of A.N. Kolmogorov. A.N. Kolmogorov works, books, papers, articles. Photographs and Portraits of A.N. Kolmogorov.
埃里克·韦斯坦因. Chapman-Kolmogorov Equation. MathWorld.

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