The maximum of branching Brownian motion in $\mathbb{R}^d$

(When the dimension d is clear from the context, we omit it from the notation, writing e.g. mt for mt(d), etc.) When d = 1, Bramson [5] proved the convergence in distribution of maxv∈NtX (v) t − mt(1), and the limit was identified by Lalley and Selke [10] to be the limit of a certain derivative martingale. It is not hard to deduce from their results and methods (see, e.g., [15, Thm. 1.1]) that, when d= 1,