Left Tail of the Derivative Martingale in a Gaussian BRW in the Entire Subcritical Regime

Xinxin Chen; Yichao Huang; Heng Ma

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arXiv · 2508.11983

Left Tail of the Derivative Martingale in a Gaussian BRW in the Entire Subcritical Regime

Xinxin Chen·Yichao Huang·Heng Ma

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Abstract

We establish a rather sharp two-sided estimate for the left tail probability of the derivative martingale limit in a binary Gaussian branching random walk throughout the entire subcritical regime, confirming a conjecture by Lacoin, Rhodes, and Vargas (\emph{Duke Math. J.} 171(3):483--545, 2022) in the case of Gaussian multiplicative cascades.

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Xinxin Chen, Yichao Huang, Heng Ma. 2025-08-16. Left Tail of the Derivative Martingale in a Gaussian BRW in the Entire Subcritical Regime. https://arxiv.org/abs/2508.11983

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