We consider the minimum of a super-critical branching random walk. Addario-Berry and Reed [Ann. Probab. 37 (2009) 1044–1079] proved the tightness of the minimum centered around its mean value. We show that a convergence in law holds, giving the analog of a well-known result of Bramson [Mem. Amer. Math. Soc. 44 (1983) iv+190] in the case of the branching Brownian motion.
"Convergence in law of the minimum of a branching random walk." Ann. Probab. 41 (3A) 1362 - 1426, May 2013. https://doi.org/10.1214/12-AOP750