# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.