On $L^p$-convergence of the Biggins martingale with complex parameter

Alexander Iksanov; Xingang Liang; Quansheng Liu

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arXiv · 1903.00524

On $L^p$-convergence of the Biggins martingale with complex parameter

Alexander Iksanov·Xingang Liang·Quansheng Liu

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Abstract

We prove necessary and sufficient conditions for the $L^p$-convergence, $p>1$, of the Biggins martingale with complex parameter in the supercritical branching random walk. The results and their proofs are much more involved (especially in the case $p\in (1,2)$) than those for the Biggins martingale with real parameter. Our conditions are ultimate in the case $p\geq 2$ only.

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Alexander Iksanov, Xingang Liang, Quansheng Liu. 2019-03-01. On $L^p$-convergence of the Biggins martingale with complex parameter. https://arxiv.org/abs/1903.00524

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