Convergence of discrete snakes (Q2576794)

The limit of discrete snakes is studied under conditions less stringent than those of \textit{P. Chassaing} and \textit{G. Schaeffer} [Probab. Theory Relat. Fields 128, No. 2, 161--212 (2004; Zbl 1041.60008)], \textit{B. Gittenberger} [J. Theor. Probab. 16, No. 4, 1063--1067 (2003; Zbl 1050.60080)], or \textit{J.-F. Marckert} and \textit{A. Mokkadem} [Ann. Probab. 31, No. 3, 1655--1678 (2003; Zbl 1049.05026)]. When the increments \(Y\) are centered and \({\mathbf P}(|Y|> y)= o(y^{-4})\), then the suitably normalized discrete snake converges weakly to the Brownian snake. When \(Y\) has a heavier tail, displacements occur which are too large for the limit to be continuous. Moreover, if the \(Y\) are not centered, a drift appears. In general, the limit is a combination of three competing parts: a drift, a Brownian snake, and a random set of jumps (``jumping snake''). In any case, if \(Y\) is centered with non-vanishing finite variance, the occupation measure of the discrete snake converges to the integrated super-Brownian excursion. The proofs rely on the convergence of the codings of the discrete snake, using ``tours'' associated with the snake.