Generalization of Cramér's and Linnik's factorization theorems in the continuation theory of distribution functions (Q1068430)

Let \(F_ 1\) and \(F_ 2\) be two non-degenerate distribution functions such that (1) \(F_ 1*F_ 2(x)=\Phi_{a,\sigma}*P_{\lambda}(x),\) \(x\leq x_ 0\), where \(x_ 0\) is a fixed number, \(\Phi_{a,\sigma}\) is the normal distribution function with the parameters a, \(\sigma^ 2>0\) and \(P_{\lambda}\) is the Poisson distribution function with the parameter \(\lambda\geq 0.\) If the characteristic functions of \(F_ 1\) and \(F_ 2\) have no zeros in the upper half plane (we assume that they are analytical there) then \[ F_ i=\Phi_{a_ i,\sigma_ i}*P_{\lambda_ i},\quad i=1,2,\quad where\quad a_ 1+a_ 2=a,\quad \sigma^ 2_ 1+\sigma^ 2_ 2=\sigma^ 2,\quad \lambda_ 1+\lambda_ 2=\lambda. \] This assertion generalizes the results of H. Cramer and Yu. V. Linnik [see \textit{E. Lukacs}, Characteristic functions. 2nd ed. (1970; Zbl 0201.204)]. In the paper there are some analogous results where the condition (1) is replaced by \(| F_ 1*F_ 2(x)- \Phi_{a,\sigma}*P_{\lambda}(x)| =O(e^{-cx^ 2})\) for some \(c>\sigma^{-2}\).