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Boundaries from Inhomogeneous Bernoulli Trials

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Random Walks, Boundaries and Spectra

Part of the book series: Progress in Probability ((PRPR,volume 64))

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Abstract

The boundary problem is considered for inhomogeneous increasing random walks on the square lattice \(\mathbb{Z}^2_+\) with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles.

Mathematics Subject Classification (2000). Primary 60J10, 60J50.

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References

  1. D. Aldous, Exchangeability and related topics. L. Notes Math. 1117, Springer, Berlin, 1985.

    Google Scholar 

  2. D. Aldous and J. Pitman, On the zero-one law for exchangeable events. Ann. Prob. (1979), 704–723.

    Google Scholar 

  3. R. Arratia, A. Barbour and S. Tavar´e, Logarithmic Combinatorial Structures: A Probabilistic Approach. European Math. Soc., 2003.

    Google Scholar 

  4. C. Charalambides, Combinatorial Methods in Discrete Distributions. Wiley, 2004.

    Google Scholar 

  5. B.M. Baker and D.E. Handelman, Positive polynomials and time-dependent integervalued random variables. Canadian J. Math. 44 (1992), 3–41.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Flajolet, P. Dumas and V. Puyhaubert, Some exactly solvable models in urn process theory. DMTCS proc. AG (2006), 58–118.

    Google Scholar 

  7. G. Fayolle, R. Iasnogorodski and V. Malyshev, Random walks in the quarter-plane. Springer 1999.

    Google Scholar 

  8. S.B. Frick, M. Keane, K. Petersen and I.A. Salama, Ergodicity of the adic transformation on the Euler graph. Math. Proc. Camb. Phil. Soc. 141 (2006), 231–238.

    Article  MathSciNet  Google Scholar 

  9. S.B. Frick and K. Petersen, Random permutations and unique fully supported ergodicity for the Euler adic transformation. Ann. IHP – Prob. Stat. 2008.

    Google Scholar 

  10. A. Gnedin, Coherent random permutations with record statistics. DMTCS proc. AH (2007) 147–158.

    Google Scholar 

  11. A. Gnedin, A species sampling model with finitely many types. Elec. Comm. Probab. 2010.

    Google Scholar 

  12. A. Gnedin and G. Olshanski, Coherent random permutations with descent statistic and the boundary problem for the graph of zigzag diagrams. Intern. Math. Res. Notes Article ID 51968 (2006).

    Google Scholar 

  13. A. Gnedin and G. Olshanski, The boundary of the Eulerian number triangle. Moscow Math. J. 6 (2006), 461–475.

    MathSciNet  MATH  Google Scholar 

  14. A. Gnedin and G. Olshanski, A q-analogue of de Finetti’s theorem. Elec. J. Combinatorics (2009) paper R78.

    Google Scholar 

  15. A. Gnedin and G. Olshanski, q-Exchangeability via quasi-invariance. Ann. Probab. (2010).

    Google Scholar 

  16. A. Gnedin and J. Pitman,Gibbs exchangeable partitions and Stirling triangles. J. Math. Sci. 138 (2006), 5674–5685.

    Article  MathSciNet  Google Scholar 

  17. K.R. Goodearl, Partially ordered abelian groups with interpolation. Mathematical Surveys and Monographs, number 20, American Mathematical Society, Providence, R.I., 1986.

    Google Scholar 

  18. F. Hausdorff, Summationsmethoden und Momentfolgen I. Math. Z. 9 (1921), 74–109.

    MathSciNet  Google Scholar 

  19. I. Ignatiouk-Robert and C. Loree, Martin boundary of a killed random walk on a quadrant. http://arxiv.org/abs/0903.0070, 2009.

  20. S. Kerov. Combinatorial examples in the theory of AF-algebras. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 172 (Differentsialnaya Geom. Gruppy Li i Mekh. Vol. 10) (1989) 55–67, 1989. Translated in J. Soviet Math. (1992) 59(5), 1063–1071.

    Google Scholar 

  21. S. Kerov, A. Okounkov and G. Olshanski, The boundary of Young graph with Jack edge multiplicities. Intern. Math. Res. Notices (1998) 173–199.

    Google Scholar 

  22. J. Mineka, A criterion for tail events for sums of independent random variables. Prob. Th. Rel. Fields (1973) 25 163–170.

    MathSciNet  MATH  Google Scholar 

  23. S. Orey, Tail events for sums of independent random variables, J. Math. Mech. (1966) 15 937–951.

    MathSciNet  MATH  Google Scholar 

  24. K. Petersen and A. Varchenko, The Euler adic dynamical system and path counts in the Euler graph. http://arxiv.org/abs/0811.1733 (2009).

  25. J. Pitman, Combinatorial stochastic processes, Springer L. Notes Math., vol. 1875, 2006.

    Google Scholar 

  26. J. Pitman, An extension of de Finetti’s theorem Adv. Appl. Probab. (1978) 10 268–270.

    Article  Google Scholar 

  27. G. Strasser, Generalisations of the Euler adic (2010) Preprint.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Mathematical Institute, Utrecht University, 80 010, NL-3508, TA Utrecht, The Netherlands

    Alexander Gnedin

Authors
  1. Alexander Gnedin
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Correspondence to Alexander Gnedin .

Editor information

Editors and Affiliations

  1. Mathematisches Inst., Universität Jena, Ernst-Abbe-Platz 2, Jena, 07737, Germany

    Daniel Lenz

  2. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Florian Sobieczky

  3. Inst. Mathematik C, TU Graz, Steyrergasse 30, Graz, 8010, Austria

    Wolfgang Woess

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Gnedin, A. (2011). Boundaries from Inhomogeneous Bernoulli Trials. In: Lenz, D., Sobieczky, F., Woess, W. (eds) Random Walks, Boundaries and Spectra. Progress in Probability, vol 64. Springer, Basel. https://doi.org/10.1007/978-3-0346-0244-0_6

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