Asymptotic expansions in the compound Poisson limit theorem (Q1969260)

Consider a triangular array \(X_{nj}\), \(j=1,\dots, m_n\), \(n\geq 1\), of independent integer-valued random variables. Let \(p_{nj}(k)= P\{X_{nj}= k\}\), \(k= 0,\pm 1,\dots\), \(F_n(x)= P\{X_n\leq x\}\), \(x\in \mathbb{R}\), where \(X_n= \sum_{j=1}^{m_n} X_{nj}\). It is well known that if \(\lim_{n\to\infty} \max_{1\leq j\leq m_n} (1-p_{nj} (0))= 0\), then the weak closure of \(\{F_n (\cdot)\), \(n\geq 1\}\) is made up of compound Poisson laws \(G(\cdot; \{\pi\})\) with characteristic function \(\varphi(t)= \exp \{\sum_{k\neq 0} (e^{itk}- 1)\pi(k)\}\), where \(\pi(k)\geq 0\) and \(\sum_{k\neq 0} \pi(k)< \infty\). The author derives asymptotic expansions of \(R_n(x)= F_n(x)- G(x; (\widehat{\pi}_n))\) for suitable sequences \((\widehat{\pi}_n(k))_{n\geq 1}\), \(k\neq 0\), with \(\widehat{\pi}_n(k)\geq 0\) and \(\sum_{k\neq 0} \widehat{\pi}_n(k)< \infty\). The results obtained generalize previous results of the author [Litov. Mat. Sb. 2, No. 1, 35-48 (1962; Zbl 0128.38004)] for the case of the Poisson limit law.