Suppose that we are given a time series where consecutive samples are believed to come from a probabilistic source, that the source changes from time to time and that the total number of sources is fixed. Our objective is to estimate the distributions of the sources. A standard approach to this problem is to model the data as a hidden Markov model (HMM). However, since the data often lacks the Markov or the stationarity properties of an HMM, one can ask whether this approach is still suitable or perhaps another approach is required. In this paper we show that a maximum likelihood HMM estimator can be used to approximate the source distributions in a much larger class of models than HMMs. Specifically, we propose a natural and fairly general non-stationary model of the data, where the only restriction is that the sources do not change too often. Our main result shows that for this model, a maximum-likelihood HMM estimator produces the correct second moment of the data, and the results can be extended to higher moments.
Let X_ 1 ,X_ 2 , be independent identically distributed random variables taking values in a finite set X and consider the conditional joint distribution of the first m elements of the sample X_ 1 , , X_ n on the condition that X_ 1 =x_ 1 and the sliding block sample average of a function h( , ) defined on X^ 2 exceeds a threshold > Eh(X_ 1 , X_ 2 ) . For m fixed and n , this conditional joint distribution is shown to converge m the m -step joint distribution of a Markov chain started in x_ 1 which is closest to X_ l , X_ 2 , in Kullback-Leibler information divergence among all Markov chains whose two-dimensional stationary distribution P( , ) satisfies P(x, y)h(x, y) , provided some distribution P on X_ 2 having equal marginals does satisfy this constraint with strict inequality. Similar conditional limit theorems are obtained when X_ 1 , X_ 2 , is an arbitrary finite-order Markov chain and more general conditioning is allowed.