Extremum of a time-inhomogeneous branching random walk

Abstract

Consider a time-inhomogeneous branching random walk, generated by the point process L_n which composed by two independent parts: ‘branching’offspring X_n with the mean \mathrm{1}+{B(\mathrm{1}+n)}^{-\beta } for \beta \in (\mathrm{0},\mathrm{1}) and ‘displacement’ {{\displaystyle \xi }}_n with a drift {A(\mathrm{1}+n)}^{-\mathrm{2}\alpha } for \alpha \in (\mathrm{0},\mathrm{1}/\mathrm{2}), where the ‘branching’ process is supercritical for B>0 but ‘asymptotically critical’ and the drift of the ‘displacement’ {{\displaystyle \xi }}_n is strictly positive or negative for \left|A\right|\rangle \mathrm{0} but ‘asymptotically’ goes to zero as time goes to infinity. We find that the limit behavior of the minimal (or maximal) position of the branching random walk is sensitive to the ‘asymptotical’ parameter \beta and \alpha.

Keywords

Branching random walk / time-inhomogeneous / branching process / random walk

References

[1] Addario-Berry L, Reed B. Minima in branching random walks. Ann Probab, 2009, 37: 1044–1079

[2] Aïdékon E. Convergence in law of the minimum of a branching random walk. Ann Probab, 2013, 41: 1362–1426

[3] Biggins J D. The first- and last-birth problems for a multitype age-dependent branching process. Adv Appl Probab, 1976, 8(3): 446–459

[4] Biggins J D. Chernoff's Theorem in the branching random walk. J Appl Probab, 1977, 14: 630–636

[5] Bramson M D. Minimal displacement of branching random walk. Probab Theory Related Fields, 1978, 45: 89–108

[6] Csáki E, Földes A, Révész P. Transient nearest neighbor random walk on the line. J Theoret Probab, 2009, 22: 100–122

[7] Csáki E, Földes A, Révész P. Transient nearest neighbor random walk and Bessel process. J Theoret Probab, 2009, 22: 992–1009

[8] Csáki E, Földes A, Révész P. On the number of cutpoints of the transient nearest neighbor random walk on the line. J Theoret Probab, 2010, 23: 624–638

[9] Dembo A, Zeitouni O. Large Deviations Techniques and Applications.Berlin: Springer, 1998

[10] D’Souza J C, Biggins J D. The supercritical Galton-Watson process in varying environments. Stochastic Process Appl, 1992, 42: 39–47

[11] Fang M, Zeitouni O. Branching random walks in time-inhomogeneous environments. Electron J Probab, 2012, 17(67): 18

[12] Fujimagari T. On the extinction time distribution of a branching process in varying environments. Adv Appl Probab, 1980, 12: 350–366

[13] Hammersley J M. Postulates for subadditive processes. Ann Probab, 1974, 2: 652–680

[14] Hong W M, Yang H. Cutoff phenomenon for nearest Lampertis random walk. Methodol Comput Appl Probab, 2019, 21(4): 1215–1228

[15] Hu Y, Shi Z. Minimal position and critical martingale convergence in branching random walks, and directed polymers on disordered trees. Ann Probab, 2009, 2: 742–789

[16] Kingman J F C. The first birth problem for age-dependent branching process. Ann Probab, 1975, 3: 790–801

[17] Lamperti J. Criteria for the recurrence or transience of stochastic processes. I. J Math Anal Appl, 1960, 1: 314–330

[18] Lamperti J. Criteria for stochastic processes. II. Passage-time moments. J Math Anal Appl, 1963, 7: 127–145

[19] Lyons R, Pemantle R, Peres Y. Conceptual proofs of Llog L criteria for mean behavior of branching processes. Ann Probab, 1995, 23: 1125–1138

[20] Mallein B. Maximal displacement in a branching random walk through interfaces. Electron J Probab, 2015, 20: 1–40

[21] Mallein B. Maximal displacement of a branching random walk in time-inhomogeneous environment. Stochastic Process Appl, 2015, 125: 3958–4019

[22] Mallein B, Milos P. Maximal displacement of a supercritical branching random walk in a time-inhomogeneous random environment. Stochastic Process Appl, 2019, 129: 3239C–3260

[23] Mcdiarmid C. Minimal positions in a branching random walk. Ann Appl Probab, 1995, 5: 128–139

[24] Shi Z. Random walks and trees. ESAIM Proc, 2011, 31: 1{39

[25] Zhang X, Hou W, Hong W. Limit theorems for the minimal position of a branching random walk in random environment. Markov Processes Related Fields, 2020, 26: 839–860

RIGHTS & PERMISSIONS

Higher Education Press