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In the traditional framework of spectral learning of stochastic time seriesmodels, model parameters are estimated based on trajectories of fully recordedobservations. However, real-world time series data often contain missingvalues, and worse, the distributions of missingness events over time are oftennot independent of the visible process. Recently, a spectral OOM learningalgorithm for time series with missing data was introduced and proved to beconsistent, albeit under quite strong conditions. Here we refine the algorithmand prove that the original strong conditions can be very much relaxed. Wevalidate our theoretical findings by numerical experiments, showing that thealgorithm can consistently handle missingness patterns whose dynamic interactswith the visible process.

A Consistent Method for Learning OOMs from Asymptotically Stationary Time Series Data Containing Missing Values