{'fraction_non_alphanumeric': 0.017789072426937738, 'fraction_numerical': 0.0, 'mean_word_length': 5.912280701754386, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Some asymptotic results for near critical branching processes

Near critical single type Bienaymé-Galton-Watson (BGW) processes are considered. It is shown that, under appropriate conditions, Yaglom distributions of suitably scaled BGW processes converge to that of the corresponding diffusion approximation. Convergences of stationary distributions for Q-processes and models with immigration to the corresponding distributions of the associated diffusion approximations are established as well. Although most of the work is concerned with the single type case, similar results for multitype settings can be obtained. As an illustration, convergence of Yaglom distributions of suitably scaled multitype subcritical BGW processes to that of the associated diffusion model is established.