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Suppose that we are given a time series where consecutive samples arebelieved to come from a probabilistic source, that the source changes from timeto time and that the total number of sources is fixed. Our objective is toestimate the distributions of the sources. A standard approach to this problemis to model the data as a hidden Markov model (HMM). However, since the dataoften lacks the Markov or the stationarity properties of an HMM, one can askwhether this approach is still suitable or perhaps another approach isrequired. In this paper we show that a maximum likelihood HMM estimator can beused to approximate the source distributions in a much larger class of modelsthan HMMs. Specifically, we propose a natural and fairly general non-stationarymodel of the data, where the only restriction is that the sources do not changetoo often. Our main result shows that for this model, a maximum-likelihood HMMestimator produces the correct second moment of the data, and the results canbe extended to higher moments.

Clustering Time Series and the Surprising Robustness of HMMs