  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.
