Abstract
Consider the Markov Branching Process with continuous time. Our focus is on the limit properties of transition functions of this process. Using differential analogue of the Basic Lemma we prove local limit theorems for all cases and observe invariant properties of considering process.
Highlights
Introduction and PreliminariesLet the random function Z(t), t ∈ T = [0; +∞), be the population size at the moment t of monotype individuals that are capable to perish and transform into individuals of random number of the same type
Each individual existing at epoch t independently of his history and of each other for a small time interval (t; t + ε) transforms into j ∈ N0 \ {1} individuals with probability ajε + o(ε) and with probability 1 + a1ε + o(ε) each individual survives or makes evenly one descendant, where N0 = {0} ∪ {N = 1, 2, . . .} and {aj} represent intensities of individuals’ transformation where aj ≥ 0 for j ∈ N0 \ {1} and 0 < a0 < −a1 = ∑j∈N0\{1} aj < ∞
Aforesaid population process Z(t) describes the branching scheme of population of individuals in which the intensity of transformation is independent of population size and of time
Summary
For studying of evolution of MBP Z(t) is suffice to set the transition functions P1j(t) These probabilities in turn, as it has been noted, can be calculated using the local densities {aj} by relation. Long-term properties of process Z(t) are investigated on non-zero trajectories that is conditioned to event T > t In this context we state some classical results on asymptote of conditioned distribution P{∗} = P{∗ | T > t} derived first by Chistyakov [8] and Sevastyanov [4]. In noncritical case (Theorem 13) we obtain an exponential invariance property of MBP Z(t) and discuss the criteria for the λZ-classification of its state space, where λZ is the decay parameter. We refine Theorems A and D consolidating them in the Theorem 16 under minimal moment conditions
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