Remarks on infinitely divisible approximations to the binomial law (Q2769665)

For any signed measure of bounded variation \(W\) concentrated on integers denote NEWLINE\[NEWLINE|W|_r= \Bigl(\sum_{k\in\mathbb{Z}} \bigl|W\{k\}\bigr |^r \Bigr)^{1/r}, 1\leq r<\infty, \quad\text{and}\quad |W |_\infty =\sup_{k\in\mathbb{Z}} \bigl|W\{k\}\bigr |.NEWLINE\]NEWLINE Let \(H_n={\mathcal L} (\xi_1+ \cdots+ \xi_n)\), where \(\xi_j\), \(j=1,\dots,n\), are independent, \(P \{\xi_j=1\} =p_j=1-P\{\xi_j=0)= 1-q_j\). Under assumptions that \(\sum^n_{j=1} p_jq_j\geq 1\) and \(p_j\leq C_0<1\), \(j=1,\dots,n\), the infinitely divisible distribution \(D_n\) concentrated on integers is constructed such that for \(r\geq 1\) NEWLINE\[NEWLINE|H_n-D_n|_r\leq C(r)\left( \sum^n_{j=1} p^2_j+1\right) \left( \sum^n_{j=1} p_jq_j \right)^{(1-5r)/2r} NEWLINE\]NEWLINE and NEWLINE\[NEWLINE|H_n-D_n|_\infty\leq C\left( \sum^n_{j=1} p^2_j+1\right) \left(\sum^n_{j=1} p_jq_j\right)^{-5/2}.NEWLINE\]NEWLINE Asymptotic expansions are also discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].