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

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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… Click to show full 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 Aidekon 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.

Keywords: proof seneta; seneta heyde; optimal assumptions; random walks; walks optimal; branching random

Journal Title: Electronic Journal of Probability
Year Published: 2019

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