Moments, large and moderate deviations for branching random walks with immigration in random environments

We consider an Rd-valued branching random walk with immigration in an i.i.d. time-dependent environment. For t is an element of Rd, let Zn(t) be the partition function of the system, and Wn(t) be the intrinsic sub-martingale formed by the normalization of Zn(t). By decomposing the family tree and using sums to characterise the upper bounds of the moments of Biggins's martingale, we study the change rates of the moments EWn(t)s for s is an element of R, and give sufficient conditions for the finiteness of supn EWn(t)s. Based on these moment results, large and moderate deviation principles are established for log Zn(t).(c) 2023 Elsevier Inc. All rights reserved.