On application of slowly varying functions with remainder in the theory of Galton–Watson branching process

We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Galton–Watson branching processes. Consider the critical case so that the generating function of the per-capita offspring distribution has the infinite second moment, but its tail is regularly varying with remainder. We improve the Basic Lemma of the theory of critical Galton-Watson branching processes and refine some well-known limit results.