一般随机串的马尔可夫嵌入
Markovian embeddings of general random strings
摘要：
设A是有限集，X是A值随机变量序列.我们不假设这些随机变量之间有任何特定的相关结构;特别是，X可能是非马尔可夫序列。X的一个适应嵌入是一个形式为R（X_1），R（X_1，X_2），R（X_1，X_2，X_3）等的序列，其中R是定义在有限长序列上的一个变换。在这个扩展的摘要中，我们刻画了一个广泛的适应嵌入的X，导致一阶齐次马尔可夫链。证明了任意变换R在这类中有唯一的粗加细R'，使得R'（X_1，X_2，X_3），R '（X_1，X_2，X_2）等是马尔可夫的. (By加细意味着R '（u）= R'（v）蕴涵R（u）=R（v），而粗加细意味着R'是我们变换类中R的任何其他加细的确定性函数。我们提出了一个特定的嵌入，我们表示为R^X，它特别适合于分析X中正则表达式描述的模式的出现。对一个0和1的非马尔可夫序列的玩具例子进行了深入的分析：建立了X_1，…，X_n，当n趋于无穷大时，而高斯渐近分布则适用于另一种正则模式。
英文摘要：
Let A be a finite set and X a sequence of A-valued random variables. We do not assume any particular correlation structure between these random variables; in particular, X may be a non-Markovian sequence. An adapted embedding of X is a sequence of the form R(X_1), R(X_1,X_2), R(X_1,X_2,X_3), etc where R is a transformation defined over finite length sequences. In this extended abstract we characterize a wide class of adapted embeddings of X that result in a first-order homogeneous Markov chain. We show that any transformation R has a unique coarsest refinement R' in this class such that R'(X_1), R'(X_1,X_2), R'(X_1,X_2,X_3), etc is Markovian. (By refinement we mean that R'(u)=R'(v) implies R(u)=R(v), and by coarsest refinement we mean that R' is a deterministic function of any other refinement of R in our class of transformations.) We propose a specific embedding that we denote as R^X which is particularly amenable for analyzing the occurrence of patterns described by regular expressions in X. A toy example of a non-Markovian sequence of 0's and 1's is analyzed thoroughly: discrete asymptotic distributions are established for the number of occurrences of a certain regular pattern in X_1,...,X_n, as n tends to infinity, whereas a Gaussian asymptotic distribution is shown to apply for another regular pattern.
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