Generalised autoregressive processes

2022-01-10 — 2023-08-11

Wherein generalisations of autoregressive processes are presented, the linear AR(1) special case is examined, and variance, correlation and conditional distribution properties are derived.

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

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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.
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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.
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