  In this article, we introduce parallel-in-time methods for state and
parameter estimation in general nonlinear non-Gaussian state-space models using
the statistical linear regression and the iterated statistical posterior
linearization paradigms. We also reformulate the proposed methods in a
square-root form, resulting in improved numerical stability while preserving
the parallelization capabilities. We then leverage the fixed-point structure of
our methods to perform likelihood-based parameter estimation in logarithmic
time with respect to the number of observations. Finally, we demonstrate the
practical performance of the methodology with numerical experiments run on a
graphics processing unit (GPU).
