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

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.

## Classic Gaussian

### Discrete time

Given a \(K \times K\) real matrix \(\Phi\) with all the eignevalues of \(\Phi\) in to the interval \((-1,1)\), and given a sequence \(\varepsilon_t\) of multivariate normal variables \(\varepsilon_t \sim \mathrm{N}(0, \Sigma)\), with \(\boldsymbol{\Sigma}\) a \(K \times K\) positive definite symmetric real matrix, the stationary distribution of the process \[ \mathbf{x}_t=\varepsilon_t+\boldsymbol{\Phi} \mathbf{x}_{t-1}=\sum_{h=0}^t \boldsymbol{\Phi}^h \varepsilon_{t-h} \quad ? \] is given by the Lyapunov equation, or just by basic variance identities. It is Gaussian with \(\mathcal{N}(0, \Lambda)\) where the following recurrence relation holds for \(\Lambda\), \[ \Lambda=\Phi \mathbf{x} \Phi^{\top}+\Sigma. \] The solution is also, apparently, the limit of a summation \[ \Lambda=\sum_{k=0}^{\infty} \Phi^k \Sigma\left(\Phi^{\top}\right)^k. \]

### Continuous time

Suppose we use a Wiener process \(W\) as the driving noise in continuous time with some small increment \(\epsilon\), \[ d \mathbf{x}(t)=-\epsilon A \mathbf{x}(t) d t+ \epsilon B d W(t) \] This is the Ornstein-Uhlenbeck process. If stable, at stationarity it an analytic stationary density \(\mathbf{x}\sim\mathcal{N}(0, \Lambda)\) where \[ \Lambda A+A \Lambda =\epsilon B B^{\top}. \]

## Gamma

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

*Proceedings of the 29th International Coference on International Conference on Machine Learning*, 1771–78. ICML’12. Madison, WI, USA: Omnipress.

*Proceedings of the 39th International Conference on Machine Learning*, 414–34. PMLR.

*Proceedings of the 31st International Conference on Machine Learning*, 1683–91. Beijing, China: PMLR.

*JMLR*, April.

*SIAM Review*58 (3): 377–441.

*arXiv:2106.00087 [Math]*, May.

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