  This note aims at presenting several new theoretical results for the compound
Poisson point process, which follows the work of Zhang \emph{et al.}
[Insurance~Math.~Econom.~59(2014), 325-336]. The first part provides a new
characterization for a discrete compound Poisson point process (proposed by
{Acz{\'e}l} [Acta~Math.~Hungar.~3(3)(1952), 219-224]), it extends the
characterization of the Poisson point process given by Copeland and Regan
[Ann.~Math.~(1936): 357-362]. Next, we derive some concentration inequalities
for discrete compound Poisson point process (negative binomial random variable
with unknown dispersion is a significant example). These concentration
inequalities are potentially useful in count data regressions. We give an
application in the weighted Lasso penalized negative binomial regression whose
KKT conditions of penalized likelihood hold with high probability and then we
derive non-asymptotic oracle inequalities for a weighted Lasso estimator.
