Generalised autoregressive processes



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

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

References

Barndorff-Nielsen, O. E. 2001. Superposition of Ornstein–Uhlenbeck Type Processes.” Theory of Probability & Its Applications 45 (2): 175–94.
Barndorff-Nielsen, Ole Eiler, and Robert Stelzer. 2011. Multivariate supOU Processes.” The Annals of Applied Probability 21 (1): 140–82.
Barndorff-Nielsen, Ole E., and Neil 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) 63 (2): 167–241.
Foti, Nicholas J., and Sinead Williamson. 2012. A Survey of Non-Exchangeable Priors for Bayesian Nonparametric Models.” arXiv:1211.4798 [Cs, Stat], November.
Griffin, J E. n.d. “Time-Dependent Stick-Breaking Processes,” 34.
Griffin, J.E. 2011. The Ornstein–Uhlenbeck Dirichlet Process and Other Time-Varying Processes for Bayesian Nonparametric Inference.” Journal of Statistical Planning and Inference 141 (11): 3648–64.
Grunwald, G K, R J Hyndman, and L M Tedesco. n.d. “A Unified View of Linear AR(1) Models,” 26.
Pigorsch, Christian, and Robert Stelzer. 2009. On the Definition, Stationary Distribution and Second Order Structure of Positive Semidefinite Ornstein–Uhlenbeck Type Processes.” Bernoulli 15 (3): 754–73.
Wolpert, Robert L. 2021. Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math], May.
Wolpert, Robert L., and Lawrence D. Brown. 2021. Markov Infinitely-Divisible Stationary Time-Reversible Integer-Valued Processes.” arXiv:2105.14591 [Math], May.

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