Stationary Gamma processes

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 require 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 $\mathsf{g}(t)\sim \mathrm{Gamma}(\alpha ,\lambda )$ univariate marginal distributions and autocorrelation function ${\rho }^{|s-t|}$ for $\mathsf{g}(s),\mathsf{g}(t)$. Each will be defined on some time index set $\mathcal{T}$, either $\mathcal{T}=\mathsf{z}$ 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 $AR(1)$ process in discrete time, or the Ornstein-Uhlenbeck process in continuous time), and there is just one $AR(1)$-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.

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)