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Asymptotical local probabilities of lower deviations for branching process in random environment with geometric distributions of descendants

  • Konstantin Yu. Denisov EMAIL logo
Published/Copyright: October 12, 2022

Abstract

We consider local probabilities of lower deviations for branching process Zn=Xn,1++Xn,Zn1 in random environment η. We assume that η is a sequence of independent identically distributed random variables and for fixed environment η the distributions of variables Xi,j are geometric ones.We suppose that the associated random walk Sn=ξ1++ξn has positive mean μ and satisfies left-hand Cramer’s condition Eexp(hξi)< if h<h<0 for some h<1. Under these assumptions, we find the asymptotic representation of local probabilities P(Zn= exp(θn) ) for θ[ θ1,θ2 ](μ;μ) for some non-negative μ.


Note

Originally published in Diskretnaya Matematika (2020) 32,№3, 24–37 (in Russian).


Funding statement: This work was supported by the Russian Science Foundation (project 19-11-001115) in Steklov Mathematical Institute of Russian Academy of Sciences.

Acknowledgment

The author is grateful to A. V. Shklyaev for constant attention and useful discussions and also to the anonimous reviewer for a number of valuable corrections.

References

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Received: 2020-05-28
Published Online: 2022-10-12
Published in Print: 2022-10-26

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