Poisson approximation in \(\chi^2\) distance by the Stein-Chen approach (Q2692549)

Let \(I_1,\ldots,I_n\) be independent Bernoulli random variables, where \(I_j\) has mean \(p_j\) and \(\Theta=\max_{1\leq j\leq n}p_j\). Denoting \(S_n=I_1+\cdots+I_n\) and \(\lambda=\mathbb{E}S_n\), the main result of the present paper is an upper bound on the \(\chi^2\) distance between \(S_n\) and a Poisson random variable of the same mean, defined by \[ \chi^2(S_n,\mathcal{P}(\lambda))=\sum_{m\geq0}\left|\frac{\mathbb{P}(S_n=m)}{e^{-\lambda}\frac{\lambda^m}{m!}}-1\right|^2e^{-\lambda}\frac{\lambda^m}{m!}\,. \] In particular, the author shows that if \(\Theta^2e^\Theta<1\), then \[ \chi^2(S_n,\mathcal{P}(\lambda))\leq\frac{1}{2}\left(\frac{\lambda_2}{\lambda}\right)^2\Theta^2\frac{2e^\Theta-1}{1-\Theta^2e^\Theta}e^\Theta+\frac{\lambda_2\lambda_3}{\lambda^2}e^\Theta+\frac{1}{2}\left(\frac{\lambda_2}{\lambda}\right)^2\,, \] where \(\lambda_k=p_1^k+\cdots+p_n^k\). This bound is comparable to others available in the literature; its novelty comes in its relatively short proof, which is an application of the Stein-Chen method in conjunction with the Charlier-Parseval identity. This should help open the door to further applications of the Stein-Chen method for approximation in \(\chi^2\) distance, for example to Poisson approximation for sums of dependent Bernoulli random variables.