Stationary Gamma processes

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.

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

See matrix and vector Gamma processes.

References