## Seneta-Heyde norming in the branching random walk.(English)Zbl 0873.60062

Summary: In the discrete-time supercritical branching random walk, there is a Kesten-Stigum type result for the martingales formed by the Laplace transform of the $$n$$th generation positions. Roughly, this says that for suitable values of the argument of the Laplace transform the martingales converge in mean provided an “$$X\log X$$” condition holds. Here it is established that when this moment condition fails, so that the martingale converges to zero, it is possible to find a (Seneta-Heyde) renormalization of the martingale that converges (in probability) to a finite nonzero limit when the process survives. As part of the proof, a Seneta-Heyde renormalization of the general (Crump-Mode-Jagers) branching process is obtained; in this case the convergence holds almost surely. The results rely heavily on a detailed study of the functional equation that the Laplace transform of the limit must satisfy.

### MSC:

 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60G50 Sums of independent random variables; random walks
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### References:

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