Tail estimates for martingale under "LLN" norming sequence

Seneta-Heyde norming for branching random walks with $α$-stable spine

We consider branching random walks with a spine in the domain of attraction of an $α$-stable Lévy process. For this process, the classical derivative martingale in general degenerates in the limit. We first determine the quantity replacing the derivative martingale and show that it converges to a non-degenerate limit under a certain LlogL-type condition which we assume to be optimal. We go on to give the Seneta-Heyde norming for the critical additive martingale under the same assumptions. The proofs are based on the methods introduced in our previous paper which considered the finite variance …

Exponential Bounds in the Law of Iterated Logarithm for Martingales

In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norming in the spirit of the classical Law of Iterated Logarithm. Key words: Martingales, exponential estimations, moment, Banach spaces of random variables, tail of distribution, conditional expectation.

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

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 Aïdékon 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 b…

大-$\theta$ Ewens-Pitman 模型的鞅方法

A Martingale Approach to Large-$θ$ Ewens-Pitman Model

本文用鞅方法重新证明大-$\theta$ Ewens-Pitman 模型中部分数的 LLN 与 CLT，获得更短证明、更优收敛速率及 Berry-Esseen 界，并填补 $\alpha\in(0,1)$ 的空白。