Notes on Discrete Compound Poisson Point Process and Its Concentration Inequalities

The first part of this notes provides a new characterization for discrete compound Poisson point process (proposed by {Acz{\'e}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.