Critical branching processes evolving in a unfavorable random environment
-
Vladimir A. Vatutin
and
Elena E. Dyakonova
Abstract
Let {Zn, n = 0, 1, 2, …} be a critical branching process in a random environment, and {Sn, n = 0, 1, 2, …} be its associated random walk. It is known that if the increments of this random walk belong (without centering) to the domain of attraction of a stable law, then there exists a regularly varying at infinity sequence a1, a2, … such that conditional distributions
converge weakly to the distribution of strictly positive proper random variable. In this paper we add to this result the description of the asymptotic behavior of the probability
where φ (n) → ∞ for n → ∞ in such a way that φ (n) = o(an).
Originally published in Diskretnaya Matematika (2022) 34, №3, 20–33 (in Russian).
Funding statement: This work was supported by the Russian Science Foundation under grant №19-11-00111, https://rscf.ru/project/19-11-00111/.
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Articles in the same Issue
- Frontmatter
- On a relationship between linear and differential characteristics of binary vector spaces mappings and diffusion characteristics over blocks of imprimitivity systems of translation group of the binary vector space
- Methods of linear and differential relations in cryptography
- On total irregular labelings with no-hole weights of some planar graphs
- Critical branching processes evolving in a unfavorable random environment
Articles in the same Issue
- Frontmatter
- On a relationship between linear and differential characteristics of binary vector spaces mappings and diffusion characteristics over blocks of imprimitivity systems of translation group of the binary vector space
- Methods of linear and differential relations in cryptography
- On total irregular labelings with no-hole weights of some planar graphs
- Critical branching processes evolving in a unfavorable random environment
