# A revisited proof of the Seneta-Heyde norming for branching random walks under optimal assumptions

## TL;DR

This paper presents a simplified proof of the Seneta-Heyde norming for branching random walks' critical additive martingale, using novel tools and estimates under optimal assumptions.

## Contribution

It introduces new methods that replace second moment estimates with truncated first moments and employs explicit potential kernel expressions for random walks.

## Key findings

- Simplified proof of Seneta-Heyde norming for branching random walks.
- Development of tools replacing second moment estimates with first moment bounds.
- Explicit expression for the potential kernel of killed random walks.

## Abstract

We introduce a set of tools which simplify and streamline the proofs of limit theorems concerning near-critical particles in branching random walks under optimal assumptions. We exemplify our method by giving another proof of the Seneta-Heyde norming for the critical additive martingale, initially due to A\"id\'ekon and Shi. The method involves in particular the replacement of certain second moment estimates by truncated first moment bounds, and the replacement of ballot-type theorems for random walks by estimates coming from an explicit expression for the potential kernel of random walks killed below the origin. Of independent interest might be a short, self-contained proof of this expression, as well as a criterion for convergence in probability of non-negative random variables in terms of conditional Laplace transforms.