## Electronic Communications in Probability

### Convergence of complex martingales in the branching random walk: the boundary

#### Abstract

Biggins [Uniform convergence of martingales in the branching random walk. Ann. Probab., 20(1):137–151, 1992] proved local uniform convergence of additive martingales in $d$-dimensional supercritical branching random walks at complex parameters $\lambda$ from an open set $\Lambda \subseteq \mathbb{C} ^d$. We investigate the martingales corresponding to parameters from the boundary $\partial \Lambda$ of $\Lambda$. The boundary can be decomposed into several parts. We demonstrate by means of an example that there may be a part of the boundary, on which the martingales do not exist. Where the martingales exist, they may diverge, vanish in the limit or converge to a non-degenerate limit. We provide mild sufficient conditions for each of these three types of limiting behaviors to occur. The arguments that give convergence to a non-degenerate limit also apply in $\Lambda$ and require weaker moment assumptions than the ones used by Biggins.

#### Article information

Source
Electron. Commun. Probab., Volume 22 (2017), paper no. 18, 14 pp.

Dates
Accepted: 16 February 2017
First available in Project Euclid: 1 March 2017

https://projecteuclid.org/euclid.ecp/1488337251

Digital Object Identifier
doi:10.1214/17-ECP50

Mathematical Reviews number (MathSciNet)
MR3615669

Zentralblatt MATH identifier
1359.60105

#### Citation

Kolesko, Konrad; Meiners, Matthias. Convergence of complex martingales in the branching random walk: the boundary. Electron. Commun. Probab. 22 (2017), paper no. 18, 14 pp. doi:10.1214/17-ECP50. https://projecteuclid.org/euclid.ecp/1488337251

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