First-order autoregressive models for gamma and exponential processes

In this paper we propose an autoregressive representation for a particular type of stationary Gamma( θ –1 , v ) process whose n -dimensional joint distributions have Laplace transform | I n + θS n V n | –v , where S n = diag( s 1 , · ··, s n ), V n is an n × n positive definite matrix with elements υ ij = p |i–j|i 2 , i, j = 1, ···, n and p is the lag-1 autocorrelation of the gamma process. We also generalize the two-parameter NEAR(1) model of Lawrance and Lewis (1981) to an exponential first-order autoregressive model with three parameters. The correlation structure and higher-order properties of the two proposed models are also given.