Convergence of Discrete Snakes
Convergence of Discrete Snakes
复制标题
离散蛇的收敛
DOI：
10.1007/s10959-005-7252-9
复制
发表时间：
2005
影响因子：
0.8
通讯作者：
J. Marckert
中科院分区：
文献类型：
--
作者：
S. Janson;J. Marckert
摘要离散蛇是一种利用条件Galton-Watson树和随机ID构造的树状结构。在本文中，我们证明了如果
$$\Mathbb{E}Y=0$$和
$$\mathbb{P}(|Y|>；y)=o(y^{-4})$$，则离散蛇弱收敛于布朗蛇(这一结果在假设下是已知的
$$\mathbb{E}Y^{8+\varepsilon}<；+\inty$$)。此外，如果这个条件不成立，并且Y的尾巴足够正则，我们证明了离散的蛇弱收敛到一个我们称为跳蛇的对象。在这两种情况下，占领测度的极限都是积分的超布朗游程。证明依赖于离散Snake编码的收敛，并借助于称为巡回的两个过程。
AbstractThe discrete snake is an arborescent structure built with the help of a conditioned Galton-Watson tree and random i.i.d. increments Y. In this paper, we show that if
$$\mathbb{E}Y= 0$$ and
$$\mathbb{P}(| Y| > y)= o(y^{-4})$$, then the discrete snake converges weakly to the Brownian snake (this result was known under the hypothesis
$$\mathbb{E}Y^{8+\varepsilon} < +\infty$$). Moreover, if this condition fails, and the tails of Y are sufficiently regular, we show that the discrete snake converges weakly to an object that we name jumping snake. In both case, the limit of the occupation measure is shown to be the integrated super-Brownian excursion. The proofs rely on the convergence of the codings of discrete snake with the help of two processes, called tours.