  In the traditional framework of spectral learning of stochastic time series
models, model parameters are estimated based on trajectories of fully recorded
observations. However, real-world time series data often contain missing
values, and worse, the distributions of missingness events over time are often
not independent of the visible process. Recently, a spectral OOM learning
algorithm for time series with missing data was introduced and proved to be
consistent, albeit under quite strong conditions. Here we refine the algorithm
and prove that the original strong conditions can be very much relaxed. We
validate our theoretical findings by numerical experiments, showing that the
algorithm can consistently handle missingness patterns whose dynamic interacts
with the visible process.
