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Small deviations of a Galton-Watson process with immigration

Sidorova, N; (2018) Small deviations of a Galton-Watson process with immigration. Bernoulli , 24 (4B) pp. 3494-3521. 10.3150/17-BEJ967. Green open access

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Abstract

We consider a Galton–Watson process with immigration (Zn), with offspring probabilities (pi) and immigration probabilities (qi). In the case when p0 = 0, p1 =/ 0, q0 = 0 (that is, when essinf(Zn) grows linearly in n), we establish the asymptotics of the left tail P{W < ε}, as ε ↓ 0, of the martingale limit W of the process (Zn). Further, we consider the first generation K such that ZK > essinf(ZK) and study the asymptotic behaviour of K conditionally on {W < ε}, as ε ↓ 0. We find the growth scale and the fluctuations of K and compare the results with those for standard Galton–Watson processes.

Type: Article
Title: Small deviations of a Galton-Watson process with immigration
Open access status: An open access version is available from UCL Discovery
DOI: 10.3150/17-BEJ967
Publisher version: http://doi.org/10.3150/17-BEJ967
Language: English
Additional information: This version is the version of record. For information on re-use, please refer to the publisher’s terms and conditions.
Keywords: conditioning; Galton–Watson processes; Galton–Watson trees; immigration; large deviations; lower tail; martingale limit; small value probabilities
UCL classification: UCL
UCL > Provost and Vice Provost Offices
UCL > Provost and Vice Provost Offices > UCL BEAMS
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences
UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics
URI: https://discovery.ucl.ac.uk/id/eprint/1520058
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