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Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

  • Elisabeth Bauernschubert (a1)

Abstract

We establish recurrence and transience criteria for critical branching processes in random environments with immigration. These results are then applied to the recurrence and transience of a recurrent random walk in a random environment on ℤ disturbed by cookies inducing a drift to the right of strength 1.

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Copyright

Corresponding author

Postal address: Mathematisches Institut, Eberhard Karls Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany. Email address: elisabeth.bauernschubert@uni-tuebingen.de

References

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[1] Babillot, M., Bougerol, P. and Elie, L. (1997). The random difference equation X n =A n X n−1+B n in the critical case. Ann. Prob. 25, 478493.
[2] Basdevant, A.-L. and Singh, A. (2008). On the speed of a cookie random walk. Prob. Theory Relat. Fields 141, 625645.
[3] Basdevant, A.-L. and Singh, A. (2008). Rate of growth of a transient cookie random walk. Electron. J. Prob. 13, 811851.
[4] Bauernschubert, E. (2013). Perturbing transient random walk in a random environment with cookies of maximal strength. Ann. Inst. H. Poincaré Prob. Statist. 49, 638653.
[5] Baum, L. E. and Katz, M. (1965). Convergence rates in the law of large numbers. Trans. Amer. Math. Soc. 120, 108123.
[6] Benjamini, I. and Wilson, D. B. (2003). Excited random walk. Electron. Commun. Prob. 8, 8692.
[7] Élie, L. (1982). Comportement asymptotique du noyau potentiel sur les groupes de Lie. Ann. Sci. École Norm. Sup. (4) 15, 257364.
[8] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd edn. John Wiley, New York.
[9] Goldie, C. M. and Maller, R. A. (2000). Stability of perpetuities. Ann. Prob. 28, 11951218.
[10] Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.
[11] Key, E. S. (1987). Limiting distributions and regeneration times for multitype branching processes with immigration in a random environment. Ann. Prob. 15, 344353.
[12] Kosygina, E. and Zerner, M. P. W. (2008). Positively and negatively excited random walks on integers, with branching processes. Electron. J. Prob. 13, 19521952.
[13] Kosygina, E. and Zerner, M. P. W. (2013). Excited random walks: results, methods, open problems. Bull. Inst. Math. Acad. Sin. (N. S.) 8, 105157.
[14] Lukacs, E. (1975). Stochastic Convergence, 2nd edn. Academic Press, New York.
[15] Peigné, M. and Woess, W. (2011). Stochastic dynamical systems with weak contractivity properties I. Strong and local contractivity. Colloq. Math. 125, 3154.
[16] Roitershtein, A. (2007). A note on multitype branching processes with immigration in a random environment. Ann. Prob. 35, 15731592.
[17] Solomon, F. (1975). Random walks in a random environment. Ann. Prob. 3, 131.
[18] Vatutin, V. A. and Zubkov, A. M. (1993). Branching processes. II. J. Soviet Math. 67, 34073485.
[19] Vatutin, V. A., Dyakonova, E. E. and Sagitov, S. (2013). Evolution of branching processes in a random environment. Proc. Steklov Inst. Math. 282, 220242.
[20] Zerner, M. P. W. (2002). Integrability of infinite weighted sums of heavy-tailed i.i.d. random variables. Stoch. Process. Appl. 99, 8194.
[21] Zerner, M. P. W. (2005). Multi-excited random walks on integers. Prob. Theory Relat. Fields 133, 98122.
[22] Zerner, M. P. W. (2006). Recurrence and transience of excited random walks on Z d and strips. Electron. Commun. Prob. 11, 118128.

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Recurrence and Transience of Critical Branching Processes in Random Environment with Immigration and an Application to Excited Random Walks

  • Elisabeth Bauernschubert (a1)

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