Generalised Ornstein-Uhlenbeck processes

AR(1)-like processes



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Ornstein-Uhlenbeck-type autoregressive, stationary stochastic processes, e.g. stationary gamma processes, classic Gaussian noise Ornstein-Uhlenbeck processes… There is a family of such induced by every Lévy process via its bridge.

Over at Gamma processes, Wolpert (2021) notes several example constructions which “look like” Ornstein-Uhlenbeck processes, in that they are stationary-autoregressive, but constructed by different means. Should we look at processes like those here?

For fixed \(\alpha, \beta>0\) these notes present six different stationary time series, each with Gamma \(X_{t} \sim \operatorname{Ga}(\alpha, \beta)\) univariate marginal distributions and autocorrelation function \(\rho^{|s-t|}\) for \(X_{s}, X_{t} .\) Each will be defined on some time index set \(\mathcal{T}\), either \(\mathcal{T}=\mathbb{Z}\) or \(\mathcal{T}=\mathbb{R}\)

Five of the six constructions can be applied to other Infinitely Divisible (ID) distributions as well, both continuous ones (normal, \(\alpha\)-stable, etc.) and discrete (Poisson, negative binomial, etc). For specifically the Poisson and Gaussian distributions, all but one of them (the Markov change-point construction) coincide— essentially, there is just one “AR(1)-like” Gaussian process (namely, the \(\operatorname{AR}(1)\) process in discrete time, or the Ornstein-Uhlenbeck process in continuous time), and there is just one \(\operatorname{AR}(1)\)-like Poisson process. For other ID distributions, however, and in particular for the Gamma, each of these constructions yields a process with the same univariate marginal distributions and the same autocorrelation but with different joint distributions at three or more times.

References

Wolpert, Robert L. 2021. Lecture Notes on Stationary Gamma Processes.” arXiv:2106.00087 [Math], May.

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