Abstract
The 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.
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References
Aldous D. (1991). The continuum random tree. II: An overview. Stochastic analysis, Proc Symp., Durham/UK 1990, Lond. Math. Soc. Lect. Note Ser. 167, 23-70
D. Aldous (1993) ArticleTitleThe continuum random tree. III Ann. Prohab. 21 IssueID1 248–289
D. Aldous (1993) ArticleTitleTree-based models for random distribution of mass J. Statist. Phys. 73 625–641 Occurrence Handle10.1007/BF01054343
P. Billingsley (1968) Convergence of Probability Measures Wiley New York
N.H. Bingham C.M Goldie J.L. Teugels (1987) Regular Variation. Encyclopedia of Mathematics and its Applications, 27 Cambridge University Press Cambridge
P. Chassaing G. Schaeffer (2002) ArticleTitleRandom planar lattices and integrated superBrownian execursion Prob. Theory Rel. Fields. 128 IssueID2 161–212 Occurrence Handle10.1007/s00440-003-0297-8
L. Devroye et al. (1998) Branching processes and their applications in the analysis of tree structures and tree algorithms M. Habib (Eds) Probabilistic Methods for Algorithmic Discrete Mathematics, Algorithms Combin. Vol. 16. Springer Berlin 249–314
Duquesne T. Gall Le F. J. (2002) Random trees, Lèvy processes and spatial branching processes. Astèrisque 281. Soc Math. de France Paris
R. Durrett H. Kesten E. Waymire (1991) ArticleTitleOn weighted heights of random trees J Theor. Probab. 4 IssueID1 223–237 Occurrence Handle10.1007/BF01047004
A. Dvoretzky Th. Motzkin (1947) ArticleTitleA problem of arrangements Duke Math. J. 14 305–313 Occurrence Handle10.1215/S0012-7094-47-01423-3
Gittenberger B. (2003). A note on “State spaces of the snake and its tour – Convergence of the discrete snake” In Marckert, J.-F, and Mokkadem, A., (eds.), J. Theo. Probab. 16(4) 1063-1067
O. Kallenberg (2002) Foundations of Modern Probability. Probability and Its Applications EditionNumber2 Springer New York, NY
H. Kesten (1994) A limit theorem for weighted branching process trees. The Dynkin Festschrift. Markov processes and their applications ed. Freidlin Birkhäuser 153–166
H. Kesten (1995) ArticleTitleBranching random walk with a critical branching part J. Theo. Prob 8 IssueID4 921–962
Le Gall J.F. (1999). Spatial branching processes, random snakes and partial differential equations. Lectures in Mathematics, Birkhäuser
J.F. Marckert (2003) ArticleTitleThe rotation correspondence is asymptotically a dilation, Random Struct Algorithms 24 IssueID2 118–132
J.F. Marckert A. Mokkadem (2003) ArticleTitleThe depth first processes of Galton-Watson trees converge to the same Brownian excursion Ann. Probab. 31 IssueID3 1655–1678 Occurrence Handle10.1214/aop/1055425793
J.F. Marckert A. Mokkadem (2002) ArticleTitleStates spaces of the snake and of its tour - Convergence of the discrete snake J. Theor. Probab. 16 IssueID4 1015–1046 Occurrence Handle10.1023/B:JOTP.0000012004.73051.f3
V.V. Petrov (1995) Limit Theorems of Probability Theory. Sequences of Independent Random Variables. Oxford Studies in Probability 4 Oxford University Press New York
Slade, G. (1999). Lattice trees, percolation and super-Brownian motion. In Bramson M. et al. (eds.), Perplexing Problems in Probability, Festschrift in Honor of Harry Kesten, Prog Probab. 44, 35-51
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Janson, S., Marckert, JF. Convergence of Discrete Snakes. J Theor Probab 18, 615–645 (2005). https://doi.org/10.1007/s10959-005-7252-9
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DOI: https://doi.org/10.1007/s10959-005-7252-9