{
  "id": "2403.19239",
  "version": "v1",
  "published": "2024-03-28T08:52:30.000Z",
  "updated": "2024-03-28T08:52:30.000Z",
  "title": "Fluctuations of the additive martingales related to super-Brownian motion",
  "authors": [
    "Ting Yang"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $(W_{t}(\\lambda))_{t\\ge 0}$, parametrized by $\\lambda\\in\\mathbb{R}$, be the additive martingale related to a supercritical super-Brownian motion on the real line and let $W_{\\infty}(\\lambda)$ 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 $W_{\\infty}(\\lambda)-W_{t}(\\lambda)$, 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 $\\psi$ around $0$. In fact, we prove that, in the case of small $|\\lambda|$, when $\\psi$ is secondly differentiable at $0$, the limit laws are scale mixtures of the standard normal laws, and when $\\psi$ 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 $|\\lambda|$. In the latter case, we show that the fluctuations and limit laws are determined by the limiting extremal process of the super-Brownian motion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-28T08:52:30.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "super-brownian motion",
      "additive martingale",
      "limit laws",
      "fluctuations",
      "branching mechanism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}