On the maximal displacement of critical branching random walk

We consider a branching random walk initiated by a single particle at location 0 in which particles alternately reproduce according to the law of a Galton-Watson process and disperse according to the law of a driftless random walk on the integers. When the offspring distribution has mean 1 the branching process is critical, and therefore dies out with probability 1 . We prove that if the particle jump distribution has mean zero, positive finite variance η 2 , and finite 4 + ε moment, and if the offspring distribution has positive variance σ 2 and finite third moment then the distribution of the rightmost position M reached by a particle of the branching random walk satisfies P { M ≥ x } ∼ 6 η 2 / ( σ 2 x 2 ) as x → ∞ . We also prove a conditional limit theorem for the distribution of the rightmost particle location at time n given that the process survives for n generations.