Large deviation estimates for branching random walks
ESAIM: Probability and Statistics, Tome 23 (2019), pp. 823-840.

For the branching random walk drifting to −∞ we study large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem.

Reçu le :
Accepté le :
DOI : 10.1051/ps/2019006
Classification : 60F10, 60G50, 60J80
Mots-clés : Branching random walk, random walk, large deviations, first passage time
Buraczewski, Dariusz 1 ; Maślanka, Mariusz 1

1 
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     pages = {823--840},
     publisher = {EDP-Sciences},
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     zbl = {1506.60038},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2019006/}
}
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Buraczewski, Dariusz; Maślanka, Mariusz. Large deviation estimates for branching random walks. ESAIM: Probability and Statistics, Tome 23 (2019), pp. 823-840. doi : 10.1051/ps/2019006. http://www.numdam.org/articles/10.1051/ps/2019006/

[1] L. Addario-Berry and B. Reed, Minima in branching random walks. Ann. Probab. 37 (2009) 1044–1079. | DOI | MR | Zbl

[2] V.I. Afanasyev, On the maximum of a subcritical branching process in a random environment. Stochastic Process. Appl. 93 (2001) 87–107. | DOI | MR | Zbl

[3] V.I. Afanasyev, High level subcritical branching processes in a random environment. Proc. Steklov Inst. Math. 282 (2013) 4–14. | DOI | MR | Zbl

[4] E. Aïdékon, Convergence in law of the minimum of a branching random walk. Ann. Probab. 41 (2013) 1362–1426. | DOI | MR | Zbl

[5] R. Bahadur and R. Rango, Rao On deviations of the sample mean. Ann. Math. Statist. 31 (1960) 1015–1027. | DOI | MR | Zbl

[6] J.D. Biggins, The first- and last-birth problems for a multitype age-dependent branching process. Adv. Appl. Probab. 8 (1976) 446–459. | DOI | MR | Zbl

[7] D. Buraczewski, J.F. Collamore, E. Damek and J. Zienkiewicz, Large deviation estimates for exceedance times of perpetuity sequences and their dual processes. Ann. Probab. 44 (2016) 3688–3739. | DOI | MR | Zbl

[8] D. Buraczewski, E. Damek and J. Zienkiewicz, Pointwise estimates for first passage times of perpetuity sequences. Stochastic Process. Appl. 128 (2018) 2923–2951. | DOI | MR | Zbl

[9] D. Buraczewski, E. Damek and J. Zienkiewicz, Precise tail asymptotics of fixed points of the smoothing transform with general weights. Bernoulli 21 (2015) 489–504. | DOI | MR | Zbl

[10] D. Buraczewski and P. Dyszewski, Large deviation estimates for branching process in random environment. Electron. J. Probab. 23 (2018) 26 pp. | DOI

[11] D. Buraczewski and M. Maślanka, Precise large deviations for the first passage time of random walk with negative drift. Proc. Amer. Math. Soc. 147 (2019) 4045–4054. | DOI | MR | Zbl

[12] X. Chen and H. He, On large deviation probabilities for empirical distribution of branching random walks: Schröder case and Böttcher case. Preprint (2017). | arXiv

[13] A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications. Jones and Bartlett, Boston (1993). | MR | Zbl

[14] N. Gantert and T. Höfelsauer, Large deviations for the maximum of a branching random walk. Electron. Commun. Probab. 23 (2018) 34. | DOI | MR | Zbl

[15] J.M. Hammersley, Postulates for subadditive processes. Ann. Probab. 2 (1974) 652–680. | DOI | MR | Zbl

[16] T. Höglund, An Asymptotic Expression for the Probability of Ruin within Finite Time. Ann. Probab. 18 (1990) 378–389. | DOI | MR | Zbl

[17] Y. Hu and Z. Shi, Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann. Probab. 37 (2009) 742–789. | MR | Zbl

[18] P. Jelenkovic and M. Olvera-Cravioto, Maximums on trees. Stochastic Process. Appl. 125 (2015) 217–232. | DOI | MR | Zbl

[19] J.F.C. Kingman, The first birth problem for an age-dependent branching process. Ann. Probab. 3 (1975) 790–801. | MR | Zbl

[20] S. Lalley, Limit theorems for first-passage times in linear and nonlinear renewal theory. Adv. Appl. Probab. 16 (1984) 766–803. | DOI | MR | Zbl

[21] V. Petrov, On the probabilities of large deviations for sums of independent random variables. Theory Probab. Appl. 10 (1965) 287–298. | DOI | MR | Zbl

[22] A. Rouault, Precise estimates of presence probabilities in the branching random walk. Stochastic Process. Appl. 44 (1993) 27–39. | DOI | MR | Zbl

[23] Z. Shi, Branching random walks. Springer (2015). | DOI | MR

[24] B. Von Bahr Ruin probabilities expressed in terms of ladder height distributions. Scand. Actuar. J. 1974 (1974) 190–204. | DOI | MR | Zbl

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