Bivariate gamma-geometric law and its induced L\'evy process

In this article we introduce a three-parameter extension of the bivariate exponential-geometric (BEG) law (Kozubowski and Panorska, 2005). We refer to this new distribution as bivariate gamma-geometric (BGG) law. A bivariate random vector $(X,N)$ follows BGG law if $N$ has geometric distribution and $X$ may be represented (in law) as a sum of $N$ independent and identically distributed gamma variables, where these variables are independent of $N$. Statistical properties such as moment generation and characteristic functions, moments and variance-covariance matrix are provided. The marginal and conditional laws are also studied. We show that BBG distribution is infinitely divisible, just as BEG model is. Further, we provide alternative representations for the BGG distribution and show that it enjoys a geometric stability property. Maximum likelihood estimation and inference are discussed and a reparametrization is proposed in order to obtain orthogonality of the parameters. We present an application to the real data set where our model provides a better fit than BEG model. Our bivariate distribution induces a bivariate L\'evy process with correlated gamma and negative binomial processes, which extends the bivariate L\'evy motion proposed by Kozubowski et al. (2008). The marginals of our L\'evy motion are mixture of gamma and negative binomial processes and we named it ${BMixGNB}$ motion. Basic properties such as stochastic self-similarity and covariance matrix of the process are presented. The bivariate distribution at fixed time of our ${BMixGNB}$ process is also studied and some results are derived, including a discussion about maximum likelihood estimation and inference.