# Critical Bellman-Harris branching processes with long-living particles

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## Abstract

A critical indecomposable two-type Bellman-Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter *β* ∈ (0, 1]. It is shown that, contrary to the critical indecomposable Bellman-Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment *t* is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment *t* given that the population contains particles of the first type at this moment.

## Keywords

Random Walk Survival Probability STEKLOV Institute Asymptotic Representation Tauberian Theorem## Preview

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