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

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- Purdue University
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## Abstract

Consider a branching random walk in which each particle has a random number (one or more) of offspring particles that are displaced independently of each other according to a logconcave density. Under mild additional assumptions, we obtain the following results: The minimal position in the n-th generation, adjusted by its α-quantile, converges weakly to a non-degenerate limiting distribution. This limiting distribution is a minimum velocity traveling wave. There also exists a “conditional limit” of the adjusted minimal position, which has a (Gumbel) extreme value distribution delayed by a random time-lag. Consequently, the unconditional limiting distribution is a mixture of extreme value distributions. ^

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