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

Processes with Gamma marginals.
Usually when we discuss *Gamma processes* we mean Gamma-*Lévy* processes.
Such processes have independent Gamma *increments*, much like a Wiener process has independent Gaussian increments and a Poisson process has independent Poisson increments.
Gamma processes provide the classic
subordinator models,
i.e. non-decreasing Lévy processes.

There are other processes with Gamma marginals.
Much like the Gaussian process family includes many processes with Gaussian marginals, so does the Gamma.
It has a different set of natural algebraic relations to the Gaussian process.
For example, the class of Gaussian processes is closed under addition and multiplication.
The class of Gamma processes is closed under addition and *thinning* and some other weirder operations, all of which requires little more background knowledge to understand.
It turns out there are complications with multivariate Gamma processes so those are handled separately.

Gamma distributions and processes and such crop up all over the place. See also Pólya-Gamma distribution.

OK but if a process’s marginals are “Gamma-distributed”, what does that even mean? First, go and read about Gamma distributions. From that we can construct the Lévy Gamma process which is usually what we mean when we talk about Gamma processes. However, there are many more processes that we can construct with Gamma marginals; those others are here.

THEN go and read about Beta and Dirichlet distributions. and the Gamma-Beta notebook.

Now we are ready to look at stationary dependent Gamma processes.

There are Ornstein–Uhlenbeck-type constructions for Gamma processes (Gaver and Lewis 1980). See R. L. Wolpert (2021) for a modern summary and overview of several popular alternatives.

For fixed \(\alpha, \lambda>0\) these notes present six different stationary time series, each with Gamma \(\rv{g}(t) \sim \operatorname{Gamma}(\alpha, \lambda)\) univariate marginal distributions and autocorrelation function \(\rho^{|s-t|}\) for \(\rv{g}(s), \rv{g}(t)\). Each will be defined on some time index set \(\mathcal{T}\), either \(\mathcal{T}=\mathbb{\rv{z}}\) 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 \(A R(1)\) process in discrete time, or the Ornstein-Uhlenbeck process in continuous time), and there is just one \(A R(1)\)-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.

## Thinned Autoregressive Gamma

To my mind the most natural one.

We let \[ \rv{g}(0) \sim \operatorname{Gamma}(\alpha, \lambda) \] and, for \(t \in \mathbb{N}\) set \[ \rv{g}(t):=\xi(t)+\zeta(t) \] where \[ \begin{aligned} &\xi(t):=\rv{b}(t) \cdot \rv{g}(t-1), \quad \rv{b}(t) \sim \operatorname{Beta}(\alpha \rho, \alpha \bar{\rho}) \\ &\zeta(t) \sim \operatorname{Gamma}(\alpha \bar{\rho}, \lambda) \end{aligned} \] where \(\bar{\rho}:=(1-\rho)\) and all the \(\left\{\rv{b}(t)\right\}\) and \(\left\{\zeta(t)\right\}\) are independent. Then,\(\xi(t) \sim \operatorname{Gamma}(\alpha \rho, \lambda)\) and \(\zeta(t) \sim \operatorname{Gamma}(\alpha \bar{\rho}, \lambda)\) are independent, with sum \(\rv{g}(t) \sim\) \(\operatorname{Gamma}(\alpha, \lambda)\). Thus \(\left\{\rv{g}(t)\right\}\) is a Markov process with Gamma univariate marginal distribution \(\rv{g}(t) \sim \operatorname{Gamma}(\alpha, \lambda)\), now with joint characteristic function \[ \begin{aligned} \chi(s, t) &=\mathbb{E}\exp\left(i s \rv{g}(0)+i t \rv{g}(1)\right) \\ &=\mathbb{E}\exp\left\{i s\left(\rv{g}(0)-\xi(1)\right)+i(s+t) \xi(1)+i t \zeta(1)\right\} \\ &=(1-i s / \lambda)^{-\alpha \bar{\rho}}(1-i(s+t) / \lambda)^{-\alpha \rho}(1-i t / \lambda)^{-\alpha \bar{\rho}} \end{aligned} \] Note that unlike the autoregressive construction, this characteristic function of this one is symmetric in the time arguments, and therefore the process is time-reversible. In some senses this is a “more natural” autoregressive process than the Zeta-innovation AR(1) process. For one, it is easy to imagine how to generalize this to vector autoregressive processes. For another, there is a natural generalization to continuous time (R. L. Wolpert 2021, 2.6) using the Beta process in the sense of Hjort (1990) and Thibaux and Jordan (2007).

What does this look like in practice?

```
set.seed(105)
# generate a stationary thinned autoregressive Gamma series
gamp = function(T, alpha, lambda, rho) {
g = rgamma(1, alpha, rate=lambda)
b = rbeta(T, alpha*rho, alpha*(1-rho))
zeta = rgamma(T, alpha*(1-rho), rate=lambda)
gs = numeric(T)
for (i in 1:T) {
g = b[i] * g + zeta[i]
gs[i] = g
}
gs
}
T = 10000
ts = (1:T)/100
plot(ts, gamp(T, 1.0, 0.1, 0.999),
type = "l", col = 2,
ylim = c(0, 25), ylab="", xlab = "time")
lines(ts, gamp(T, 10, 1.0, 0.999),
type = "l", col = 3)
lines(ts, gamp(T, 100, 10.0, 0.999),
type = "l", col = 4)
legend("topright",
c("lambda=0.1", "lambda=1", "lambda=10"),
lty = 1, col = 2:4)
```

## Additive Zeta innovations

Fix \(0 \leq \rho<1\). Let \(\rv{g}(0) \sim \operatorname{Gamma}(\alpha, \lambda)\) and for \(t \in \mathbb{N}\) define \(\rv{g}(t)\) recursively by \[ \rv{g}(t):=\rho \rv{g}(t-1)+\zeta(t) \] for iid \(\left\{\zeta(t)\right\}\) (see Zeta distribution). The process \(\left\{\rv{g}(t)\right\}\) has Gamma univariate marginal distribution \(\rv{g}(t) \sim \operatorname{Gamma}(\alpha, \lambda)\) for every \(t \in \mathbb{R}_{+}\) and, at consecutive times \(s,t\) joint characteristic function \[ \begin{aligned} \chi(s, t) &=\operatorname{E} \exp \left(i s \rv{g}(0)+i t \rv{g}(1)\right) \\ &=\operatorname{E} \exp \left(i(s+\rho t) \rv{g}(0)+i t \zeta(1)\right) \\ &=\left[\frac{(1-i(s+\rho t) / \lambda)(1-i t / \lambda)}{1-i t \rho / \lambda}\right]^{-\alpha}. \end{aligned} \] Unlike Gaussian additive autoregressive processes, where the marginal and innovation processes are both Gaussian, in Gamma additive autoregressive processes the marginal is Gamma but the innovation is not (Lawrance 1982; Walker 2000). We can get a process that has a gamma innovation by the next construction instead.

Exercise: Generalise this to continuous time.

### Gamma-Zeta distribution

I don’t know a name for the distribution of the \(\zeta(t)\) RVs from earlier.
Let us go with *Gamma-Zeta*, because plain Zeta is taken.

It is easiest to describe that RV in terms of the characteristic function \(E e^{i \omega \zeta(t)}=(1-i \omega / \lambda)^{-\alpha}(1-i \rho \omega / \lambda)^{\alpha}=\left[\frac{\lambda-i \omega}{\lambda-i \rho \omega}\right]^{-\alpha}.\)

Simulating such RVs is easy via the algorithm of Walker (2000):

\[\lambda(t) \sim \operatorname{Gamma}(\alpha, 1), \quad N(t)|\lambda(t) \sim \mathrm{Po}\left(\frac{1-\rho}{\rho} \lambda(t)\right), \quad \zeta(t)| N(t) \sim \operatorname{Gamma}\left(N(t), \frac{\lambda}{\rho}\right).\]

However, this distribution does not seem to have an obvious density except as a Fourier transform. Let us set is aside for now, eh?

## Change-point gamma

Also from R. L. Wolpert (2021). What other marginals than Gamma can I construct with this?

Let \(\left\{\zeta_{n}: n \in \mathbb{Z}\right\} \stackrel{\text { iid }}{\sim} \mathrm{Ga}(\alpha, \beta)\) be iid Gamma random variables and let \(N_{t}\) be a standard Poisson process indexed by \(t \in \mathbb{R}\) (so \(N_{0}=0\) and \(\left(N_{t}-N_{s}\right) \sim \mathrm{Po}(t-s)\) for all \(-\infty<s<\) \(t<\infty\), with independent increments), and set \[ X_{t}:=\zeta_{n}, \quad n=N_{\lambda t} \] Then each \(X_{t} \sim \mathrm{Ga}(\alpha, \beta)\) and, for \(s, t \in \mathbb{R}, X_{s}\) and \(X_{t}\) are either identical (with probability \(\left.\rho^{|s-t|}\right)\) or independent- reminiscent of a Metropolis MCMC chain. The chf is \[ \begin{aligned} \chi(s, t) &=\mathrm{E} \exp \left(i s X_{0}+i t X_{1}\right) \\ &=\rho(1-i(s+t) / \beta)^{-\alpha}+\bar{\rho}(1-i s / \beta)^{-\alpha}(1-i t / \beta)^{-\alpha} \end{aligned} \] and once again the marginal distribution is \(X_{t} \sim \mathrm{Ga}(\alpha, \beta)\) and the autocorrelation function is \(\operatorname{Corr}\left(X_{s}, X_{t}\right)=\rho^{|s-t|}\).

## Matrix-valued Gamma-like processes

## References

*Computing*12 (3): 223–46.

*Communications of the ACM*25 (1): 47–54.

*Notices of the AMS*51 (11): 1336–47.

*Lévy Processes and Stochastic Calculus*. 2nd ed. Cambridge Studies in Advanced Mathematics 116. Cambridge ; New York: Cambridge University Press.

*Stochastic Simulation: Algorithms and Analysis*. 2007 edition. New York: Springer.

*Proceedings of the 35th Conference on Winter Simulation: Driving Innovation*, 319–26. WSC ’03. New Orleans, Louisiana: Winter Simulation Conference.

*Bernoulli*12 (1): 1–33.

*Advances in Applied Probability*33 (1): 160–87.

*Lévy Processes*. Cambridge Tracts in Mathematics 121. Cambridge ; New York: Cambridge University Press.

*Lectures on Probability Theory and Statistics: Ecole d’Eté de Probailités de Saint-Flour XXVII - 1997*, edited by Jean Bertoin, Fabio Martinelli, Yuval Peres, and Pierre Bernard, 1717:1–91. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg.

*Subordinators, Lévy Processes with No Negative Jumps, and Branching Processes*. University of Aarhus. Centre for Mathematical Physics and Stochastics ….

*Stochastic Processes with Applications*. Society for Industrial and Applied Mathematics.

*Generalized Gamma Convolutions and Related Classes of Distributions and Densities*. Lecture Notes in Statistics 76. New York: Springer Science & Business Media.

*arXiv:1502.03901 [Math, q-Fin]*, February.

*Exercises in Probability: A Guided Tour from Measure Theory to Random Processes, Via Conditioning*. Cambridge University Press.

*Journal of Applied Probability*17 (2): 467–80.

*Journal of the American Statistical Association*64 (325): 194–206.

*Non-uniform random variate generation*. New York: Springer.

*Advances in Applied Mathematics*20 (3): 285–99.

*Statistics and Computing*29 (1): 67–78.

*Publications of the Research Institute for Mathematical Sciences*40 (3): 669–88.

*The Annals of Statistics*2 (4): 615–29.

*The Annals of Mathematical Statistics*43 (5): 1634–43.

*Handbook of Computational Finance*, edited by Jin-Chuan Duan, Wolfgang Karl Härdle, and James E. Gentle, 61–88. Berlin, Heidelberg: Springer Berlin Heidelberg.

*Artificial Intelligence and Statistics*, 20–28.

*Advances in Applied Probability*12 (3): 727–45.

*Journal of Forecasting*25 (2): 129–52.

*Journal of Machine Learning Research*12 (32): 1185–1224.

*Student’s t-Distribution and Related Stochastic Processes*. SpringerBriefs in Statistics. Berlin, Heidelberg: Springer Berlin Heidelberg.

*Handbook of Beta Distribution and Its Applications*. Boca Raton: CRC Press.

*Theory of Stochastic Processes : With Applications to Financial Mathematics and Risk Theory*. Problem Books in Mathematics. New York: Springer New York.

*The Annals of Applied Probability*26 (1): 328–59.

*The Annals of Statistics*18 (3): 1259–94.

*Canadian Journal of Statistics*30 (2): 269–83.

*Probability Surveys*5: 346–415.

*Poisson Processes*. Clarendon Press.

*Bayesian Analysis*14 (4): 1037–73.

*Fluctuations of Lévy Processes with Applications: Introductory Lectures*. Second edition. Universitext. Heidelberg: Springer.

*Scandinavian Journal of Statistics*9 (4): 234–36.

*Proceedings of the 26th Annual International Conference on Machine Learning*, 601–8. ICML ’09. New York, NY, USA: ACM.

*Applied Stochastic Processes*. Universitext. Springer New York.

*Stochastic Environmental Research and Risk Assessment*25 (2): 235–51.

*Annals of the Institute of Statistical Mathematics*41 (2): 227–45.

*Annals of the Institute of Statistical Mathematics*34 (3): 591–97.

*Journal of Multivariate Analysis*39 (1): 135–53.

*Linear Algebra and Its Applications*, Tenth Special Issue (Part 2) on Linear Algebra and Statistics, 410 (November): 198–216.

*Annals of the Institute of Statistical Mathematics*44 (1): 97–106.

*Journal of Multivariate Analysis*175 (January): 104560.

*Annals of the Institute of Statistical Mathematics*37 (3): 541–44.

*Probability, Statistics, and Stochastic Processes*. Hoboken, N.J: Hoboken, N.J. : Wiley-Interscience.

*Journal of Multivariate Analysis*130 (September): 155–75.

*arXiv:1201.3256 [Math]*, January.

*Journal of the American Statistical Association*108 (504): 1339–49.

*Proceedings of the 22nd International Conference on Neural Information Processing Systems*, 1554–62. NIPS’09. Red Hook, NY, USA: Curran Associates Inc.

*Artificial Intelligence and Statistics*, 800–808. PMLR.

*Simulation and the Monte Carlo Method*. 3 edition. Wiley series in probability and statistics. Hoboken, New Jersey: Wiley.

*Lévy Processes and Infinitely Divisible Distributions*. Cambridge University Press.

*International Journal of Theoretical and Applied Finance*11 (01): 1–18.

*Artificial Intelligence and Statistics*, 877–85. PMLR.

*Journal of Applied Probability*25 (3): 501–9.

*Journal of Applied Probability*27 (2): 325–32.

*Engineering Probabilistic Design and Maintenance for Flood Protection*, edited by Roger Cooke, Max Mendel, and Han Vrijling, 67–75. Boston, MA: Springer US.

*Scandinavian Journal of Statistics*20 (3): 251–61.

*Infinite Divisibility of Probability Distributions on the Real Line*. Boca Raton: CRC Press.

*Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics*, 564–71. PMLR.

*Scandinavian Actuarial Journal*1977 (1): 31–40.

*Scandinavian Actuarial Journal*1977 (3): 121–48.

*2018 AIAA Non-Deterministic Approaches Conference*, January.

*Statistics & Probability Letters*80 (7): 697–705.

*Methodology and Computing in Applied Probability*12 (4): 695–729.

*Scandinavian Journal of Statistics*27 (3): 575–76.

*Engineering Probabilistic Design and Maintenance for Flood Protection*, edited by Roger Cooke, Max Mendel, and Han Vrijling, 77–83. Boston, MA: Springer US.

*Proceedings of the Twenty-Seventh Conference on Uncertainty in Artificial Intelligence*, 736–44. UAI’11. Arlington, Virginia, United States: AUAI Press.

*Biometrika*85 (2): 251–67.

*arXiv:2106.00087 [Math]*, May.

*arXiv:2105.14591 [Math]*, May.

*arXiv:1503.08542 [Cs, Stat]*, March.

*Advances in Mathematical Finance*, edited by Michael C. Fu, Robert A. Jarrow, Ju-Yi J. Yen, and Robert J. Elliott, 37–47. Applied and Numerical Harmonic Analysis. Birkhäuser Boston.

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