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Berry-Esseen bounds for self-normalized martingales. (English) Zbl 1391.60089
Summary: A Berry-Esseen bound is obtained for self-normalized martingales under the assumption of finite moments. The bound coincides with the classical Berry-Esseen bound for standardized martingales. An example is given to show the optimality of the bound. Applications to Student’s statistic and autoregressive process are also discussed.
MSC:
60G42 Martingales with discrete parameter
60F05 Central limit and other weak theorems
60E15 Inequalities; stochastic orderings
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[1] Bentkus, V; Götze, F, The Berry-Esseen bound for student’s statistic, Ann. Probab., 24, 491-503, (1996) · Zbl 0855.62009
[2] Bentkus, V; Bloznelis, M; Götze, F, A Berry-esséen bound for student’s statistic in the non-iid case, J. Theor. Probab., 9, 765-796, (1996) · Zbl 0923.60029
[3] Csörgő, M; Szyszkowicz, B; Wang, Q, Donsker’s theorem for self-normalized partial sums processes, Ann. Probab., 31, 1228-1240, (2003) · Zbl 1045.60020
[4] De la Peña, V.H., Lai, T.L., Shao, Q.M.: Self-normalized Processes: Limit Theory and Statistical Applications. Springer, New York (2009) · Zbl 1165.62071
[5] Giné, E; Götze, F; Mason, DM, When is the student \(t\)-statistic asymptotically standard normal?, Ann. Probab., 25, 1514-1531, (1997) · Zbl 0958.60023
[6] Haeusler, E, On the rate of convergence in the central limit theorem for martingales with discrete and continuous time, Ann. Probab., 16, 275-299, (1988) · Zbl 0639.60030
[7] Hall, P., Heyde, C.C.: Martingale Limit Theory and Its Applications. Academic, New York (1980) · Zbl 0462.60045
[8] Heyde, CC; Brown, BM, On the depature from normality of a certain class of martingales, Ann. Math. Statist., 41, 2161-2165, (1970) · Zbl 0225.60026
[9] Jing, BY; Shao, QM; Wang, Q, Self-normalized cramér-type large deviations for independent random variables, Ann. Probab., 31, 2167-2215, (2003) · Zbl 1051.60031
[10] Lai, TL; Wei, CZ, Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems, Ann. Statist., 10, 154-166, (1982) · Zbl 0649.62060
[11] Logan, BF; Mallows, CL; Rice, SO; Shepp, LA, Limit distributions of self-normalized sums, Ann. Probab., 1, 788-809, (1973) · Zbl 0272.60016
[12] Shao, QM, A cramér type large deviation result for student’s \(t-\)statistic, J. Theor. Probab., 12, 385-398, (1999) · Zbl 0927.60045
[13] Shao, QM; Wang, QM, Self-normalized limit theorems: a survey, Probab. Surv., 10, 69-93, (2013) · Zbl 1286.60029
[14] Shao, QM; Zhou, WX, Self-normalization: taming a wild population in a heavy-tailed world, Appl. Math. J. Chin. Univ., 32, 253-269, (2017) · Zbl 1399.60032
[15] Bahr, B; Esseen, CG, Inequalities for the \(r\)th absolute moment of a sum of random variables, \(1\le r \le 2\), Ann. Math. Statist., 36, 299-303, (1965) · Zbl 0134.36902
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