Further Improvement of the Basic Lemma of Critical Bienaymé-Galton-Watson Branching Systems and Its Applications

The paper considers the Bienaymé-Galton-Watson Branching Systems, in which the mean per-capita offspring number is equal to one and the variance is infinite. This system is called a critical one. We state and prove an alternative variant of Basic Lemma of the theory of critical Bienaymé-Galton-Watson system. The proved lemma essentially improves the corresponding result of the previous works of the authors. This assertion plays a key role in formulating the local limit theorem with explicit terms in the asymptotic expansion of local probabilities on positive trajectories of the system considered.