Some remarks on the stability of the characterization of the composed random variables (Q1423558)

This note examines a certain subclass \(N_{\varepsilon}\) of the class \(N\) of characteristic functions corresponding to the composition of two random variables. Informally, the \(N_{\varepsilon}\) subclass functions possess the following ``stability'' property: small changes in the component distribution functions lead to only small changes in the distribution function of the composed random variables. The article establishes stability conditions, estimations on the functions in the \(N_{\varepsilon}\) subclass, and applies these results to several (ordered) pairs of random variable distributions that are coming into composition: \(\langle\)Poisson, \(\varepsilon\)-exponential\(\rangle\), \(\langle\)Poisson, \(\varepsilon\)-normal\(\rangle\), \(\langle\)binomial, \(\varepsilon\)-exponential\(\rangle\) and \(\langle\)negative binomial, \(\varepsilon\)-exponential\(\rangle\).