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

Alexander Iksanov; Xingang Liang; Quansheng Liu

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

Probability 2019-03-05 v1

Authors: Alexander Iksanov , Xingang Liang , Quansheng Liu

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.

Keywords

lp spaces and inequalities

Cite

@article{arxiv.1903.00524,
  title  = {On $L^p$-convergence of the Biggins martingale with complex parameter},
  author = {Alexander Iksanov and Xingang Liang and Quansheng Liu},
  journal= {arXiv preprint arXiv:1903.00524},
  year   = {2019}
}

Comments

submitted, 16 pages

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