The consistency of the BIC Markov order estimator
BIC 马尔可夫阶数估计量的一致性
The Bayesian Information Criterion (BIC) estimates the order of a Markov chain (with finite alphabet $A$) from observation of a sample path $x_1, x_2,\dots, x_n$, as that value $k = \hat{k}$ that minimizes the sum of the negative logarithm of the $k$th order maximum likelihood and the penalty term $\frac{|A|^k(|A|-1)}{2}\log n$. We show that $\hat{k}$ equals the correct order of the chain, eventually almost surely as $n \rightarrow \infty$, thereby strengthening earlier…