On corrected Poisson approximations for sums of independent indicators

Let $S_n=I_1+\cdots +I_n$ be a sum of independent indicators $I_i$, with $p_i=\mathrm{Pr}(I_i=1)=1-\mathrm{Pr}(I_i=0)$, $i=1,\dots ,n$. It is well-known that the total variation distance between $S_n$ and $Z_{\lambda }$, where $Z_{\lambda }$ has a Poisson distribution with mean $\lambda ={\sum }_{i=1}^n{p}_i$, is typically of order ${\sum }_{i=1}^n{p}_i^2$. In the present work we propose a class of corrected Poisson approximations, which enable the second order factorial moment distance (and hence, the total variation distance) to be bounded above by a constant multiple of ${\sum }_{i=1}^n{p}_i^3$ and ${\sum }_{i=1}^n{p}_i^4$, hence improving the order of approximation.