  We investigate the scaling properties of products of the exponential of
birth--death processes with certain given marginal discrete distributions and
covariance structures. The conditions on the mean, variance and covariance
functions of the resulting cumulative processes are interpreted in terms of the
moment generating functions. We provide four illustrative examples of Poisson,
Pascal, binomial and hypergeometric distributions. We establish the
corresponding log-Poisson, log-Pascal, log-binomial and log-hypergeometric
scenarios for the limiting processes, including their R\'{e}nyi functions and
dependence properties.
