  We investigate a branching random walk where the displacements are
independent from the branching mechanism and have a stretched exponential
distribution. We describe the positions of the particles in the vicinity of the
rightmost particle in terms of point process convergence. As a consequence we
give a~new limit theorem for the position of the rightmost particle. Our
methods rely on providing precise large deviations for sums of i.i.d. random
variables with stretched exponential distributions outside the so-called one
big jump regime.
