Generalised autoregressive processes

January 10, 2022 — August 11, 2023

dynamical systems
Hilbert space
Lévy processes
probability
regression
signal processing
statistics
stochastic processes
time series

\[\renewcommand{\var}{\operatorname{Var}} \renewcommand{\corr}{\operatorname{Corr}} \renewcommand{\dd}{\mathrm{d}} \renewcommand{\bb}[1]{\mathbb{#1}} \renewcommand{\vv}[1]{\boldsymbol{#1}} \renewcommand{\rv}[1]{\mathsf{#1}} \renewcommand{\vrv}[1]{\vv{\rv{#1}}} \renewcommand{\disteq}{\stackrel{d}{=}} \renewcommand{\gvn}{\mid} \renewcommand{\Ex}{\mathbb{E}} \renewcommand{\Pr}{\mathbb{P}}\]

Some useful generalizations of autoregressive (i.e., AR(1)) processes.

Linear ones are an interesting case (G. K. Grunwald, Hyndman, and Tedesco 1996; Gary K. Grunwald et al. 2000).

Figure 1

1 References

Barndorff-Nielsen, O. E. 2001. “Superposition of Ornstein-Uhlenbeck Type Processes.” Theory of Probability & Its Applications.
Barndorff-Nielsen, Ole E., and Shephard. 2001. “Non-Gaussian Ornstein–Uhlenbeck-Based Models and Some of Their Uses in Financial Economics.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Barndorff-Nielsen, Ole Eiler, and Stelzer. 2011. “Multivariate supOU Processes.” The Annals of Applied Probability.
Foti, and Williamson. 2015. “A Survey of Non-Exchangeable Priors for Bayesian Nonparametric Models.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Griffin. 2011. “The Ornstein–Uhlenbeck Dirichlet Process and Other Time-Varying Processes for Bayesian Nonparametric Inference.” Journal of Statistical Planning and Inference.
Griffin. n.d. “Time-Dependent Stick-Breaking Processes.”
Grunwald, G K, Hyndman, and Tedesco. 1996. “A Unified View of Linear AR(1) Models.”
Grunwald, Gary K., Hyndman, Tedesco, et al. 2000. “Theory & Methods: Non-Gaussian Conditional Linear AR(1) Models.” Australian & New Zealand Journal of Statistics.
Pigorsch, and Stelzer. 2009. “On the Definition, Stationary Distribution and Second Order Structure of Positive Semidefinite Ornstein–Uhlenbeck Type Processes.” Bernoulli.
Wolpert. 2021. “Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math].
Wolpert, and Brown. 2021. “Markov Infinitely-Divisible Stationary Time-Reversible Integer-Valued Processes.” arXiv:2105.14591 [Math].