关于有限马氏过程记忆的估计
On estimating the memory for finitarily Markovian processes
摘要：
有限马尔可夫过程是那些过程$\{X_n\}_{n=-\infty}^{\infty}$，对于这些过程，存在有限的$K$（$K = K(\{X_n\}_{n=-\infty}^0$），使得给定整个过去的$X_1$的条件分布等于仅给定$\{X_n\}_{n=1-K}^0$的$X_1$的条件分布。$K$的最小值称为内存长度。我们给出了一个相当完整的分析问题普遍估计最小这样的价值$K$，无论是在向后的意义上，我们刚刚描述的，并在向前的意义上，其中一个观察连续的价值$\{X_n\}$的$n \geq 0$和要求的最小值$K$，使条件分布的$X_{n+1}$给定$\{X_i\}_{i=n-K+1}^n$是相同的条件分布的$X_{n+1}$给定$\{X_i\}_{i=-\infty}^n$。我们允许有限或可数无限的字母大小。
英文摘要：
Finitarily Markovian processes are those processes $\{X_n\}_{n=-\infty}^{\infty}$ for which there is a finite $K$ ($K = K(\{X_n\}_{n=-\infty}^0$) such that the conditional distribution of $X_1$ given the entire past is equal to the conditional distribution of $X_1$ given only $\{X_n\}_{n=1-K}^0$. The least such value of $K$ is called the memory length. We give a rather complete analysis of the problems of universally estimating the least such value of $K$, both in the backward sense that we have just described and in the forward sense, where one observes successive values of $\{X_n\}$ for $n \geq 0$ and asks for the least value $K$ such that the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=n-K+1}^n$ is the same as the conditional distribution of $X_{n+1}$ given $\{X_i\}_{i=-\infty}^n$. We allow for finite or countably infinite alphabet size.
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