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Deviation inequalities for a supercritical branching process in a random environment. (English) Zbl 1513.60127

Summary: Let \(\{Z_n, n\geq 0\}\) be a supercritical branching process in an independent and identically distributed random environment \(\xi = (\xi_n)_{n\geq 0}\). In this paper, we get some deviation inequalities for \(\ln(Z_{n+n_0}/Z_{n_0})\). And some applications are given for constructing confidence intervals.

MSC:

60K37 Processes in random environments
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60F10 Large deviations
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References:

[1] W. L. SMITH, W. E. WILKINSON. On branching processes in random environments. Ann. Math. Statist., 1969, 40: 814-827. · Zbl 0184.21103
[2] V. I. AFANASYEV, C. BÖINGHOFF, G. KERSTING, et al. Limit theorems for weakly subcritical branching processes in random environment. J. Theoret. Probab., 2012, 25(3): 703-732. · Zbl 1262.60083
[3] V. I. AFANASYEV, C. BÖINGHOFF, G. KERSTING. Conditional limit theorems for intermediately sub-critical branching processes in random environment. Ann. Inst. Henri Poincaré Probab. Stat., 2014, 50(2): 602-627. · Zbl 1290.60083
[4] V. A. VATUTIN. A Refinement of Limit Theorems for the Critical Branching Processes in Random Envi-ronment. Springer, Berlin, 2010.
[5] C. BÖINGHOFF, G. KERSTING. Upper large deviations of branching processes in a random environemnt -offspring distributions with geomertrically bounded tails. Stochastic Process. Appl., 2010, 120(10): 2064-2077. · Zbl 1198.60045
[6] V. BANSAYE, J. BERESTYCKI. Large deviations for branching processes in random environment. Markov Process. Related Fields, 2009, 15(4): 493-524. · Zbl 1193.60098
[7] Chunmao HUANG, Quansheng LIU. Moments, moderate and large deviations for a branching process in a random environment. Stochastic Process. Appl., 2012, 122(2): 522-545. · Zbl 1242.60087
[8] V. BANSAYE, C. BÖINGHOFF. Upper large deviations for branching processes in random environment with heavy tails. Electron. J. Probab., 2011, 16(69): 1900-1933. · Zbl 1245.60081
[9] M. V. KOZLOV. On large deviations of branching processes in a random environment: geometric distribution of descendants. Discrete Math. Appl., 2006, 16(2): 155-174. · Zbl 1126.60089
[10] M. NAKASHIMA. Lower deviations of branching processes in random environment with geometrical offspring distributions. Stochastic Process. Appl., 2013, 123(9): 3560-3587. · Zbl 1291.60175
[11] C. BÖINGHOFF. 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. · Zbl 1296.60231
[12] Yanqing WANG, Quansheng LIU. Limit theorems for a supercritical branching process with immigration in a random environment. Sci. China Math., 2017, 60(12): 2481-2502. · Zbl 1390.60315
[13] I. GRAMA, Quansheng LIU, M. MIQUCU. Berry-Esseen’s bound and Cramér’s large deviations for a su-percritical branching process in a random environment. Stochastic Process. Appl., 2017, 127(4): 1255-1281. · Zbl 1358.60089
[14] Xiequan FAN, Haijuan HU, Quansheng LIU. Uniform Cramér moderate deviations and Berry-Esseen bounds for a supercritical branching process in a random environment. Front. Math. China, 2020, 15(5): 891-914. · Zbl 1468.60102
[15] S. V. NAGAEV, V. VAKHTEL. Probability inequalities for a critical Galton-Watson process. Theory Probab. Appl., 2006, 20(2): 225-247. · Zbl 1331.60173
[16] V. H. DE lA PEÑA. A general class of exponential inequalities for martingales and ratios. Ann. Probab., 1999, 27(1): 537-564. · Zbl 0942.60004
[17] Xiequan FAN, I. GRAMA, Quansheng LIU. Cramér large deviation expansions for martingales under Bern-stein’s condition. Stochastic Process. Appl., 2013, 123(11): 3919-3942. · Zbl 1327.60069
[18] A. A. BOROVKOV. Estimates for the distribution of sums and maxima of sums of random variables when the Cramér condition is not satisfied. Siberian Math. J, 2000, 41(5): 811-848. · Zbl 0969.60047
[19] J. DEDECKER, P. DOUKHAN, Xiequan FAN. Deviation inequalities for separately Lipschitz functionals of composition of random functions. J. Math. Anal. Appl, 2019, 479(2): 1549-1568. · Zbl 1479.60041
[20] Xiequan FAN, I. GRAMA, Quansheng LIU. Deviation inequalities for martingales with applications. J. Math. Anal. Appl., 2017, 448(1): 538-566. · Zbl 1387.60034
[21] B. VON BAHR, C. G. ESSEEN. Inequlities for the rth absolute moment of a sum of random variables, 1 r 2. Ann. Math. Stat., 1965, 36(1): 299-303. · Zbl 0134.36902
[22] Xiequan FAN, I. GRAMA, Quansheng LIU. Hoeffding’s inequality for supermartingales. Stochastic Process. Appl., 2012, 122(10): 3545-3559. · Zbl 1267.60045
[23] E. RIO. On McDiarmid’s concentration inequality. Electron. Commun. Probab., 2013, 18(44): 1-11. · Zbl 1348.60042
[24] S. N. BERNSTEIN. The Theory of Probabilities. Leningrad, Moscow, 1946.
[25] S. V. NAGAEV. Large deviations of sums of independent random variables. Ann. Probab., 1979, 7(5): 745-789. · Zbl 0418.60033
[26] W. HOEFFDING. Probalility inequalities for sums of bounded random variables. J. Amer. Statist. Assoc., 1963, 58: 13-30. · Zbl 0127.10602
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