Local limit approximations for Lagrangian distributions
The Lagrange distribution, given by \[ P(0) = 0, \quad P(m) = {1\over m!} {d^{m - 1}\over dz^{m - 1}} (f'(z) g^ m(z)),\quad \quad z = 0,\;m = 1,2,\dots, \] with probability generating functions \(f\), \(g\), \(g(0)= 0\), studied for the case that \[ f(z) = \varphi(\lambda z)/\varphi(\lambda),\quad\quad g(z) = \psi(\mu z)/\psi(\mu), \] where \(\varphi\), \(\psi\) are power series with non-negative coefficients. Under suitable assumptions on \(\varphi\), \(\psi\), local limit theorems are derived for \(m \to \infty\) with \(\lambda\), \(\mu\) changing appropriately. These results lead to approximations for well-known special distributions of Lagrangian type.
scientific article; zbMATH DE number 729146
scientific article