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We consider a branching random walk on the line. Biggins and Kyprianou [6]proved that, in the boundary case, the associated derivative martingaleconverges almost surly to a finite nonnegative limit, whose law serves as afixed point of a smoothing transformation (Mandelbrot's cascade). In thepresent paper, we give a necessary and sufficient condition for thenon-triviality of this limit and establish a Kesten-Stigum-like result.

A necessary and sufficient condition for the non-trivial limit of the derivative martingale in a branching random walk