Integro-differential inequalities and the Poisson approximation
\textit{T. Cacoullos} and the authors [Ann. Probab. 22, No. 3, 1607-1618 (1994; Zbl 0835.60023)] used Stein's method to establish bounds on the difference in total variation between the (absolutely continuous) distribution of a random variable \(X\) and the normal distribution with the same mean and variance, expressed in terms of the quantities \[ U_X = \sup_{g \in L_2} {\text{Var} g (X) \over \text{Var} XE \{[g'(X)]^2\}} \quad \text{and} \quad J_X = \inf_{g \in L_2} {\text{Var} g (X) \over \text{Var} XE^2[g'(X)]}. \] It is the case that \(U_X \geq 1\), and \(J_X \leq 1\), equality in each case characterizing the normal distribution. Analogous results are obtained for Poisson approximation in total variation of nonnegative integer valued random variables, with \(g'\) now being replaced by \(\Delta g\) in the definitions of \(U_X\) and \(J_X\). Applications to sums of independent Bernoulli and negative binomial random variables are given.
Poisson approximation in total variation
scientific article; zbMATH DE number 918042