Small value probabilities for supercritical multitype branching processes with immigration

We consider the small value probability of a supercritical multi-type branching process with immigration (Zn,n≥0). It is well known that with a suitable normalization, ρ−nZn converges to a finite and non-negative limit, which can be written as Wt where WW is a random variable and t is a deterministic vector. Based on the large deviation results in Jones (2004) about the supercritical multitype branching processes, and techniques dealing with immigration used in Chu et al. (2014), we obtain the convergence rate of P(W≤ε)P(W≤ε) as ε→0+ε→0+, which is known as the small value probability of WW.