Lecture Notes on Stationary Gamma Processes

TL;DR

This paper discusses the existence and uniqueness of stationary AR(1) processes with Gamma and other infinitely-divisible distributions, presenting six different processes sharing the same marginal and autocorrelation properties.

Contribution

It introduces six distinct stationary AR(1) processes with Gamma marginals and explores their properties and simulation methods.

Findings

For Gamma distributions, multiple processes share the same autocorrelation.

Unique processes exist for Normal and Poisson distributions.

Six different processes with identical marginals and autocorrelation are described.

Abstract

For each $\lambda>0$ and every square-integrable infinitely-divisible (ID) distribution there exists at least one stationary stochastic process $t\mapsto X_t$ with the specified distribution for $X_1$ and with first-order autoregressive (AR(1)) structure in the sense that the autocorrelation of $X_s$ and $X_t$ is $\exp(-\lambda|s-t|)$ for all indices $s,t$. For the special case of the standard Normal distribution, the process $X_t$ is unique -- namely, the first-order autoregressive Ornstein-Uhlenbeck velocity process. The process $X_t$ is also uniquely determined if $X_1$ is accorded the unit rate Poisson distribution. For the Gamma distribution, however, $X_t$ is \emph{not} determined uniquely. In these lecture notes we describe six distinct processes with the same univariate marginal distributions and AR(1) autocorrelation function. We explore a few of their properties and describe methods of simulating their sample paths.