ON THE EMPTY BALLS OF A CRITICAL OR SUBCRITICAL BRANCHING RANDOM WALK

Let {Z_n}n≥0 be a critical or subcritical d-dimensional branching random walk started from a Poisson random measure whose intensity measure is the Lebesugue measure on Rd.Denote by R_n :=sup{u> 0:Z_n（{x∈ Rd:|x|<u}）=0} the radius of the largest empty ball centered at the origin of Z_n.In this work,we prove that after suitable renormalization,R_n converges in law to some non-degenerate distribution as n→∞.Furthermore,our work shows that the renormalization scales depend on the offspring law and the dimension of the branching random walk.This completes the results of Révész [13] for the critical binary branching Wiener process.