Geometric infinitely divisible autoregressive models

In this article, we discuss some geometric infinitely divisible (gid) random variables using the Laplace exponents which are Bernstein functions and study their properties. The distributional properties and limiting behavior of the probability densities of these gid random variables at \(0^{+}\) $0^{+}$ are studied. The autoregressive (AR) models with gid marginals are introduced. Further, the first order AR process is generalized to kth order AR process. We also provide the parameter estimation method based on conditional least square and method of moments for the introduced AR(1) process. We also apply the introduced AR(1) model with geometric inverse Gaussian marginals on the household energy usage data which provide a good fit as compared to normal AR(1) data.