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

Abstract : We consider a branching random walk on the line. Biggins and Kyprianou [6] proved that, in the boundary case, the associated derivative martingale converges almost surly to a fi nite nonnegative limit, whose law serves as a fixed point of a smoothing transformation (Mandelbrot's cascade). In the present paper, we give a necessary and su fficient condition for the non-triviality of this limit and establish a Kesten-Stigum-like result.
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  • HAL Id : hal-00951159, version 1
  • ARXIV : 1402.5864

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Xinxin Chen. A necessary and su cient condition for the non-trivial limit of the derivative martingale in a branching random walk. Advances of Applied Probability, 2015, 47 (3), pp.741-760. ⟨hal-00951159⟩

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