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

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

*arXiv:2106.00087 [Math]*, May.

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