Total variation asymptotics for sums of independent integer random variables

By A. Barbour, V. Čekanavičius. 2002

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A multivariate Berry–Esseen theorem with explicit constants (2018, methodology): “Similarly, Barbour and Čekanavičius [4] succeed to sharply estimate the error in the asymptotic expansions for integer random variables, but although the main argument is based on Stein’s method, appropriate smoothness of modifications of W is needed and derived separately: see the inequality (5.7”
Approximation of sums of conditionally independent variables by the translated Poisson distribution (2005, methodology): “In accordance with the notation of the previous section, let N :¼ E(W jN ) ¼ Nr, 2 :¼ var( N ) ¼ b2 r2; 2N :¼ var(W jN ) ¼ Ns2, º :¼ E( 2N ) ¼ as2, 2 :¼ var( 2N ) ¼ b2s4: Then, given N ¼ k, we can apply Theorem 3.1 from Barbour and Čekanavičius (2002) to W in order to bound DTV(k) and Dloc(k).”