Convergence of discrete snakes. (English) Zbl 1084.60049

Janson, Svante; Marckert, Jean-François

J. Theor. Probab. 18, No. 3, 615-645 (2005).

The limit of discrete snakes is studied under conditions less stringent than those of P. Chassaing and G. Schaeffer [Probab. Theory Relat. Fields 128, No. 2, 161–212 (2004; Zbl 1041.60008)], B. Gittenberger [J. Theor. Probab. 16, No. 4, 1063–1067 (2003; Zbl 1050.60080)], or J.-F. Marckert and 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.

Reviewer: Heinrich Hering (Rockenberg)

MSC:

60J80

Branching processes (Galton-Watson, birth-and-death, etc.)

Keywords:

branching random walk; weak convergence; Brownian snake; integrated super-Brownian excursion

Citations:

Zbl 1041.60008; Zbl 1050.60080; Zbl 1049.05026

References:

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