States Spaces of the Snake and Its Tour—Convergence of the Discrete Snake

@article{Marckert2003StatesSO,
  title={States Spaces of the Snake and Its Tour—Convergence of the Discrete Snake},
  author={Jean-François Marckert and Abdelkader Mokkadem},
  journal={Journal of Theoretical Probability},
  year={2003},
  volume={16},
  pages={1015-1046}
}
In this paper, we show that the states space of the Brownian snake and the states space of its tour are homeomorphic. We prove that the tour of the discrete snake (built on a geometrical Galton–Watson tree of size n) converges weakly to the tour of the Brownian snake. As a consequence, we obtain the weak convergence of the discrete snake to the Brownian snake. In a last part, we show the weak convergence of the “geometrical width” of the discrete snake to the one of the Brownian snake. 

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