# 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).

## 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]*.