Convergence of Discrete Snakes

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.