Fluctuations of the additive martingales related to super-Brownian motion
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917624749752320 |
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| author | Yang, Ting |
| author_facet | Yang, Ting |
| contents | Let $(W_{t}(λ))_{t\ge 0}$, parametrized by $λ\in\mathbb{R}$, be the additive martingale related to a supercritical super-Brownian motion on the real line and let $W_{\infty}(λ)$ be its limit. Under a natural condition for the martingale limit to be non-degenerate, we investigate the rate at which the martingale approaches its limit. Indeed, assuming certain moment conditions on the branching mechanism, we show that the tail martingale $W_{\infty}(λ)-W_{t}(λ)$, properly normalized, converges in distribution to a non-degenerate random variable, and we identify the limit laws. We find that, for parameters with small absolute value, the fluctuations are affected by the behaviour of the branching mechanism $ψ$ around $0$. In fact, we prove that, in the case of small $|λ|$, when $ψ$ is secondly differentiable at $0$, the limit laws are scale mixtures of the standard normal laws, and when $ψ$ is `stable-like' near $0$ in some proper sense, the limit laws are scale mixtures of the stable laws. However, the effect of the branching mechanism is limited in the case of large $|λ|$. In the latter case, we show that the fluctuations and limit laws are determined by the limiting extremal process of the super-Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_19239 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fluctuations of the additive martingales related to super-Brownian motion Yang, Ting Probability Let $(W_{t}(λ))_{t\ge 0}$, parametrized by $λ\in\mathbb{R}$, be the additive martingale related to a supercritical super-Brownian motion on the real line and let $W_{\infty}(λ)$ be its limit. Under a natural condition for the martingale limit to be non-degenerate, we investigate the rate at which the martingale approaches its limit. Indeed, assuming certain moment conditions on the branching mechanism, we show that the tail martingale $W_{\infty}(λ)-W_{t}(λ)$, properly normalized, converges in distribution to a non-degenerate random variable, and we identify the limit laws. We find that, for parameters with small absolute value, the fluctuations are affected by the behaviour of the branching mechanism $ψ$ around $0$. In fact, we prove that, in the case of small $|λ|$, when $ψ$ is secondly differentiable at $0$, the limit laws are scale mixtures of the standard normal laws, and when $ψ$ is `stable-like' near $0$ in some proper sense, the limit laws are scale mixtures of the stable laws. However, the effect of the branching mechanism is limited in the case of large $|λ|$. In the latter case, we show that the fluctuations and limit laws are determined by the limiting extremal process of the super-Brownian motion. |
| title | Fluctuations of the additive martingales related to super-Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2403.19239 |