Asymptotic expansions for generalized Poisson measures
Asymptotic expansions are obtained for distributions having the following structure: \(H = (1-p)F + pB\), where \(F \in{\mathcal F}_+\), \(B\in{\mathcal F}\), \(0 \leq p \leq c_0 < 1\), and \(\mathcal F\) (resp. \(\mathcal F_+\)) is the class of all distributions on \({\mathbb R} \) (resp. of those with nonnegative characteristic functions). Set \(\Delta_{\nu}={n\choose\nu}G^{n-\nu}(H-G)^{\nu}\) where \(G=\exp\{(H-E)-p^2(B-E)/2\}\), \(E\) is the distribution concentrated at the point \(0\), \(\nu=1,\ldots,s\) and \(s\leq [n/2]\). For instance, an upper bound is given for \(|\Delta_{\nu}|_h\) where \(h>0\) and \(|W|_h = \sup_{x}|W([x,x+h])|\) is an analogue of the Lévy concentration function for a signed measure \(W\). No moment assumptions are made. The key role belongs to the asymptotic expansion of \textit{H. Bergström} [Skand. Actuarietidskr. 1951, 1-34 (1951; Zbl 0045.07301)] and the method of triangular functions proposed by T. Arak and developed by A. Zajtsev. The results obtained can be simplified in some special cases, e.g., the author considers the distribution of a non-lattice discrete random variable. It is also mentioned that the generalized (signed) Poisson's measures used here are important for applications, especially in actuarial mathematics.
scientific article; zbMATH DE number 1296180
scientific article