Martingale convergence and the stopped branching random walk
Let \(\zeta _u\) be the position of the individual \(u\) in the branching random walk in \(R^d\). Let \(|u|\) stand for the generation of \(u.\) For \(\theta \in R^d\) define \(W_n(\theta)=\sum_{|u|=n}\exp\{-\theta \zeta _u\}/m^n(\theta)\) where \(m(\theta)={\mathbb E}\sum_{|u|=1}\exp\{-\theta \zeta _u\}\). Clearly, \(W_n(\theta)\) may be viewed as a Laplace-Stieltjes transformation of the point process \(\{\zeta _u : |u|=n\}\) weighted by its mean and, as known, the branching property of this subpopulation is crucial in proving that \(W_n(\theta)\) is a martingale. The author searches for other `subpopulations' like \(\{\zeta _u : |u|=n\}\) on which the branching property can be established and from which one could show existence of other martingales like \(W_n(\theta)\). For the cases when such martingales exist he finds the relationships between their limits and the limit \(W(\theta)=\lim_{n\to\infty}W_n(\theta).\) The main tools in construction and analysis of the mentioned martingales are the idea of stopping lines which is a bit more restrictive than that suggested by \textit{P. Jagers} [Stochastic Processes Appl. 32, No. 2, 183-212 (1989; Zbl 0678.92009)] and the method of size biased trees due to \textit{R. Lyons, R. Pemantle} and \textit{Y. Peres} [Ann. Probab. 23, No. 3, 1125-1138 (1995; Zbl 0840.60077)].
scientific article; zbMATH DE number 1454837