A nonuniform bound for the approximation of Poisson binomial by Poisson distribution (Q1415159)

Let \(X_1,X_2,\ldots,X_n\) be independent, possibly not identically distributed random variables with \(P(X_i=1)=1-P(X_i=0)=p_i\). Define \(S_n=X_1+X_2+\cdots+X_n\). Let \(\lambda=\sum_{i=1}^np_i\) and let \(\mathcal P_\lambda\) be the Poisson random variable with parameter \(\lambda\). Using Stein-Chen method, the author proves the following main results: Let \(\lambda\in(0,1)\) and \(\omega_0\in\{1,2,\ldots,n-1\}\). Then \[ \left| P(S_n=\omega_0)-P(\mathcal P_\lambda=\omega_0)\right| \leq\frac{1}{\omega_0}\sum_{i=1}^n p^2_i. \]