Advancing limit theorems in Markov branching systems through Karamata slowly varying functions

We develop an analytical framework that connects Karamata slowly varying (SV) functions with the long-time asymptotics of continuous-time Markov branching systems in the noncritical regime. Building upon Zolotarev’s classical subcritical theory, we obtain refined asymptotic representations for the bridled-survival probability and the associated generating functions under a structural expansion of the Harris–Sevastyanov transform near the fixed point, allowing both finite-variance and infinite-variance settings. A central outcome is the emergence of a slowly varying function that plays a structural role in the asymptotic representation: it governs the deviation from the principal exponential scaling and provides a functional characterization of the generalized Kolmogorov constant. Moreover, by working within subclasses of SV-functions with polynomially controlled remainders, we derive quantitative error bounds and explicit convergence-rate refinements beyond leading-order asymptotics. As an application, the same approach extends to Markov branching–immigration systems, providing an explicit rate at which their generating functions converge to the limiting distribution. Overall, the results unify slow-variation techniques with branching dynamics and yield a versatile framework for refined asymptotic analysis in a broad class of Markov branching models.