# Branching random walk: Seneta-Heyde norming

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

In Trees: Proceedings of a Workshop held in Versailles June 14-16, 1995, eds B. Chauvin, S. Cohen, A. Rouault. Birkhäuser, Basel. (1996), 31-50

1991 Mathematics Subject Classification: 60J80

## Abstract

In the discrete-time branching random walk, the martingale formed by taking the Laplace transform of the \$n\$th generation point process is known, for suitable values of the argument, to converge in \$L_{1}\$ under an \$X\log X\$ condition, and to converge to zero when this moment condition fails. This paper examines the strategy used in Biggins and Kyprianou (1996) to prove that, when the \$X\log X\$ condition fails, there exists a Seneta-Heyde renormalisation of the martingale that converges in probability to a non-trivial random variable. To bring out how the method works it is first discussed in the context of the Galton-Watson process. The paper is concluded by extending the results to the case of the continuous-time Markov branching random walk.

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