Title: Notes on Discrete Compound Poisson Point Process and Its Concentration Inequalities,
Abstract: The first part of this notes provides a new characterization for discrete
compound Poisson point process (proposed by {Acz{é}l}
[Acta~Math.~Hungar.~3(3)(1952), 219-224.]), which extends the characterization
of Poisson point process given by Copeland and Regan [Ann.~Math.~(1936):
357-362.]. Next, we derive some concentration inequalities for discrete
compound Poisson random variable and discrete compound Poisson point process
(Poisson and negative binomial are the special cases). These concentration
inequalities are potentially useful. In high-dimensional negative binomial
regression with weighted Lasso penalty, we give the application that KKT
conditions of penalized likelihood holds with high probability.