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

## TL;DR

This paper establishes precise conditions for the $L^p$-convergence of the Biggins martingale with complex parameters in supercritical branching random walks, extending previous results especially for $p$ between 1 and 2.

## Contribution

It provides necessary and sufficient conditions for $L^p$-convergence with complex parameters, advancing understanding beyond real parameter cases.

## Key findings

- Conditions are necessary and sufficient for $p>1$ convergence.
- Results are more complex for $p$ in (1,2) than for real parameters.
- Conditions are definitive for $p \\geq 2$.

## 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.