Limit theorems for the minimal position in a branching random walk with independent logconcave displacements

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of e ach other according to a logconcave density. Under mild additional assumpti ons, we obtain the following results: the minimal position in the nth gener ation, adjusted by its alpha-quantile, converges weakly to a non-degenerate limiting distribution. There also exists a 'conditional limit' of the adju sted minimal position, which has a (Gumbel) extreme value distribution dela yed by a random time-lag. Consequently, the unconditional limiting distribu tion is a mixture of extreme value distributions.