# Generalised Ornstein-Uhlenbeck processes

## AR(1)-like processes


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. arXiv:2106.00087 [Math], May.

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