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