Branching random walk with a critical branching part

We consider the branching treeT(n) of the first (n+1) generations of a critical branching process, conditioned on survival till time βn for some fixed β>0 or on extinction occurring at timekn withkn/n→β. We attach to each vertexv of this tree a random variableX(v) and define\(S(v) = \Sigma _{w \varepsilon \pi (0,v)} X(w)\), where π(0,v) is the unique path in the family tree from its root tov. FinallyMnis the maximal displacement of the branching random walkS(·), that isMn=max{S(v):v∈T(n)}. We show that if theX(v), v∈T(n), are i.i.d. with mean 0, then under some further moment conditionn−1/2Mn converges in distribution. In particular {n−1/2Mn}n⩾1 is a tight family. This is closely related to recent results about Aldous' continuum tree and Le Gall's Brownian snake.