Conditional limit theorems for general branching processes

Abstract

In this paper we generalise the so-called Yaglom conditional limit theorems to the general branching process counted by the values of a random characteristic, as suggested by Jagers (1974). Even when restricted to the special case of the usual population-size process, our results are stronger than those previously available.

Height and contour processes of Crump-Mode-Jagers forests (I): general distribution and scaling limits in the case of short edges

“…Remark 5.5. In [30], Sagitov investigated (in the non-triangular setting) the size of a CMJ process conditioned to survive at large time under the short edge assumption, i.e., when E(V * 1 ) < ∞ and E(Y * 1 ) < ∞ (see also Section 8 and Green [12]). The population size is described in the limit in terms of a continuous state branching process where space and time are scaled analogously as in Corollary 5.3.…”

Asymptotics of continuous-time discrete state space branching processes for large initial state

“…To the best of the authors knowledge this question has not been discussed rigorously in the literature for continuous-time discrete state space branching processes. Related questions for discrete-time Galton-Watson processes have been studied extensively in the literature (see for example Lamperti [27,28] or Green [19]), however in this situation time is usually scaled as well, which make these approaches different from the continuous-time case. The article of Sagitov [37] contains related results, however the critical case is considered and again an additional time scaling is used.…”