RESEARCH ARTICLE

Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment

  • Xiequan FAN , 1 ,
  • Haijuan HU 2 ,
  • Quansheng LIU 3,4
Expand
  • 1. Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
  • 2. School of Mathematics and Statistics, Northeastern University at Qinhuangdao, Qinhuangdao 066004, China
  • 3. Université de Bretagne-Sud, LMBA, UMR CNRS 6205, Campus de Tohannic, 56017 Vannes, France
  • 4. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

Received date: 11 Feb 2020

Accepted date: 10 Oct 2020

Published date: 15 Oct 2020

Copyright

2020 Higher Education Press

Abstract

Let {Zn, n0}be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(Zn+n0/Zn0 ) uniformly in n0 ,which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n0= 0. The extension is interesting in theory, and is motivated by applications. A new method is developed for the proofs; some conditions of Grama et al. are relaxed in our present setting. An example of application is given in constructing confidence intervals to estimate the criticality parameter in terms of log(Zn+n0/Zn0 ) and n.

Cite this article

Xiequan FAN , Haijuan HU , Quansheng LIU . Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment[J]. Frontiers of Mathematics in China, 2020 , 15(5) : 891 -914 . DOI: 10.1007/s11464-020-0868-3

1
Afanasyev V I, Böinghoff C, Kersting G, Vatutin V A. Conditional limit theorems for intermediately subcritical branching processes in random environment. Ann Inst Henri Poincaré Probab Stat, 2014, 50(2): 602–627

DOI

2
Athreya K B, Karlin S. On branching processes with random environments: I Extinction probabilities. Ann Math Stat, 1971, 42(5): 1499–1520

DOI

3
Athreya K B, Karlin S. Branching processes with random environments: II Limit theorems. Ann Math Stat, 1971, 42(6): 1843–1858

DOI

4
Bansaye V, Berestycki J. Large deviations for branching processes in random environment. Markov Process Related Fields, 2009, 15(4): 493–524

5
Bansaye V, Böinghoff C. Upper large deviations for branching processes in random environment with heavy tails. Electron J Probab, 2011, 16(69): 1900–1933

DOI

6
Bansaye V, Vatutin V. On the survival probability for a class of subcritical branching processes in random environment. Bernoulli, 2017, 23(1): 58–88

DOI

7
Böinghoff C. Limit theorems for strongly and intermediately supercritical branching processes in random environment with linear fractional offspring distributions. Stochastic Process Appl, 2014, 124(11): 3553–3577

DOI

8
Böinghoff C, Kersting G. Upper large deviations of branching processes in a random environment-offspring distributions with geometrically bounded tails. Stochastic Process Appl, 2010, 120: 2064–2077

DOI

9
Cramér H. Sur un nouveau théorème-limite de la théorie des probabilités. Actualités Sci Indust, 1938, 736: 5–23

10
Fan X. Cramér type moderate deviations for self-normalized ψ-mixing sequences. J Math Anal Appl, 2020, 486(2): 123902

DOI

11
Fan X, Grama I, Liu Q. Deviation inequalities for martingales with applications. J Math Anal Appl, 2017, 448(1): 538–566

DOI

12
Fan X, Grama I, Liu Q, Shao Q M. Self-normalized Cramér type moderate deviations for stationary sequences and applications. Stochastic Process Appl, 2020, 130: 5124–5148

DOI

13
Grama I, Liu Q, Miqueu M. Berry-Esseen’s bound and Cramér’s large deviations for a supercritical branching process in a random environment. Stochastic Process Appl, 2017, 127: 1255–1281

DOI

14
Huang C, Liu Q. Moments, moderate and large deviations for a branching process in a random environment. Stochastic Process Appl, 2012, 122: 522–545

DOI

15
Kozlov M V. On large deviations of branching processes in a random environment: geometric distribution of descendants. Discrete Math Appl, 2006, 16(2): 155–174

DOI

16
Linnik Y V. On the probability of large deviations for the sums of independent variables. In: Proc 4th Berkeley Sympos Math Statist and Prob, Vol 2. Berkeley: Univ California Press, 1961, 289–306

17
Nagaev S V. Large deviations of sums of independent random variables. Ann Probab, 1979, 7: 745–789

DOI

18
Nakashima M. Lower deviations of branching processes in random environment with geometrical offspring distributions. Stochastic Process Appl, 2013, 123(9): 3560–3587

DOI

19
Smith W L, Wilkinson W E. On branching processes in random environment. Ann Math Stat, 1969, 40(3): 814–827

DOI

20
Tanny D. A necessary and sufficient condition for a branching process in a random environment to grow like the product of its means. Stochastic Process Appl, 1988, 28(1): 123–139

DOI

21
Vatutin V A. A refinement of limit theorems for the critical branching processes in random environment. In: Workshop on Branching Processes and their Applications. Lect Notes Stat Proc, Vol 197. Berlin: Springer, 2010, 3–19

DOI

22
Vatutin V, Zheng X. Subcritical branching processes in random environment without Cramer condition. Stochastic Process Appl, 2012, 122: 2594–2609

DOI

23
Wang Y, Liu Q. Limit theorems for a supercritical branching process with immigration in a random environment. Sci China Math, 2017, 60(12): 2481–2502

DOI

Outlines

/