  Gaussian processes provide a flexible framework for forecasting, removing
noise, and interpreting long temporal datasets. State space modelling (Kalman
filtering) enables these non-parametric models to be deployed on long datasets
by reducing the complexity to linear in the number of data points. The
complexity is still cubic in the state dimension $m$ which is an impediment to
practical application. In certain special cases (Gaussian likelihood, regular
spacing) the GP posterior will reach a steady posterior state when the data are
very long. We leverage this and formulate an inference scheme for GPs with
general likelihoods, where inference is based on single-sweep EP (assumed
density filtering). The infinite-horizon model tackles the cubic cost in the
state dimensionality and reduces the cost in the state dimension $m$ to
$\mathcal{O}(m^2)$ per data point. The model is extended to online-learning of
hyperparameters. We show examples for large finite-length modelling problems,
and present how the method runs in real-time on a smartphone on a continuous
data stream updated at 100~Hz.
