## Bernoulli

• Bernoulli
• Volume 24, Number 4B (2018), 3494-3521.

### Small deviations of a Galton–Watson process with immigration

#### Abstract

We consider a Galton–Watson process with immigration $(\mathcal{Z}_{n})$, with offspring probabilities $(p_{i})$ and immigration probabilities $(q_{i})$. In the case when $p_{0}=0$, $p_{1}\neq0$, $q_{0}=0$ (that is, when $\operatorname{essinf}(\mathcal{Z}_{n})$ grows linearly in $n$), we establish the asymptotics of the left tail $\mathbb{P}\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow0$, of the martingale limit $\mathcal{W}$ of the process $(\mathcal{Z}_{n})$. Further, we consider the first generation $\mathcal{K}$ such that $\mathcal{Z}_{\mathcal{K}}\operatorname{essinf}(\mathcal{Z}_{\mathcal{K}})$ and study the asymptotic behaviour of $\mathcal{K}$ conditionally on $\{\mathcal{W}<\varepsilon\}$, as $\varepsilon\downarrow 0$. We find the growth scale and the fluctuations of $\mathcal{K}$ and compare the results with those for standard Galton–Watson processes.

#### Article information

Source
Bernoulli, Volume 24, Number 4B (2018), 3494-3521.

Dates
Revised: June 2017
First available in Project Euclid: 18 April 2018

https://projecteuclid.org/euclid.bj/1524038761

Digital Object Identifier
doi:10.3150/17-BEJ967

Mathematical Reviews number (MathSciNet)
MR3788180

Zentralblatt MATH identifier
06869883

#### Citation

Sidorova, Nadia. Small deviations of a Galton–Watson process with immigration. Bernoulli 24 (2018), no. 4B, 3494--3521. doi:10.3150/17-BEJ967. https://projecteuclid.org/euclid.bj/1524038761

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