On the Maximal Displacement of a 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 $\eta^{2}$, and finite $4+\varepsilon$ moment, and if the offspring distribution has positive variance $\sigma^{2}$ and finite third moment then the distribution of the rightmost position $M$ reached by a particle of the branching random walk satisfies $P\{M \geq x\}\sim 6\eta^{2}/ (\sigma^{2}x^{2})$ as $x \rightarrow \infty$. 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.