Small deviations of a Galton–Watson process with immigration

Nadia Sidorova
2018 Bernoulli  
We consider a Galton-Watson process with immigration (Z n ), with offspring probabilities (p i ) and immigration probabilities (q i ). In the case when p 0 = 0, p 1 = 0, q 0 = 0 (that is, when essinf(Z n ) 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 (Z n ). Further, we consider the first generation K such that Z K > essinf(Z K ) and study the asymptotic behaviour of K conditionally on {W < ε}, as ε ↓ 0. We find
more » ... , as ε ↓ 0. We find the growth scale and the fluctuations of K and compare the results with those for standard Galton-Watson processes.
doi:10.3150/17-bej967 fatcat:4t63jhbznffd3ma7pkbxezlyui