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

Abstract

Let \{Z_n,\mathrm{}n\ge \mathrm{0}\}be a supercritical branching process in an independent and identically distributed random environment. We prove Cramér moderate deviations and Berry-Esseen bounds for log(Z_{n+n_{\mathrm{0}}}/Z_{n_{\mathrm{0}}}) uniformly in n_{\mathrm{0}}\in \mathbb{Z},which extend the corresponding results by I. Grama, Q. Liu, and M. Miqueu [Stochastic Process. Appl., 2017, 127: 1255–1281] established for n_{\mathrm{0}}=\mathrm{}\mathrm{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(Z_{n+n_{\mathrm{0}}}/Z_{n_{\mathrm{0}}}) and n.

Keywords

Branching processes / random environment / Cramér moderate deviations / Berry-Esseen bounds

References

[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

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

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

[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

[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

[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

[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

[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

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

[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

[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

[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

[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

[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

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

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

[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

[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

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

[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

RIGHTS & PERMISSIONS

Higher Education Press