Seneta-Heyde norming in the branching random walk

by J.D.Biggins and A.E.Kyprianou ).

Ann. Probab. (1997) 25 337-360.

1991 Mathematics Subject Classification: 60J80

Abstract

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 non-zero limit when the process survives. As part of the proof a Seneta-Heyde renormalization of the general (C-M-J) 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.

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See also the companion paper
Other publications by J.D. Biggins

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