ON LARGE-DEVIATION PROBABILITIES FOR THE EMPIRICAL DISTRIBUTION OF BRANCHING RANDOM WALKS WITH HEAVY TAILS

Given a branching random walk (Z(n))(n >= 0) on R, let Z(n)(A) be the number of particles located in interval A at generation n. It is well known that under some mild conditions, Z(n)(root nA)/Z(n)(R) converges almost surely to nu(A) as n -> infinity, where. is the standard Gaussian measure. We investigate its large-deviation probabilities under the condition that the step size or offspring law has a heavy tail, i.e. a decay rate of P(Z(n)(root nA)/Z(n)(R)> p) as n -> infinity, where p epsilon (nu(A), 1). Our results complete those in Chen and He (2019) and Louidor and Perkins (2015).