Berry–Esseen Bounds for Self-Normalized Martingales

Abstract

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.

Keywords

Self-normalized process / Berry–Esseen bounds / Martingales / Student’s statistic / Autoregressive process

References

[1] Bentkus V, Götze F. The Berry–Esseen bound for student’s statistic. Ann. Probab.. 1996, 24 1 491-503

[2] Bentkus V, Bloznelis M, Götze F. A Berry–Esséen bound for student’s statistic in the non-iid case. J. Theor. Probab.. 1996, 9 3 765-796

[3] Csörgő M, Szyszkowicz B, Wang Q. Donsker’s theorem for self-normalized partial sums processes. Ann. Probab.. 2003, 31 1228-1240

[4] De la Peña VH, Lai TL, Shao QM. Self-normalized Processes: Limit Theory and Statistical Applications. 2009 New York: Springer

[5] Giné E, Götze F, Mason DM. When is the student $t$-statistic asymptotically standard normal?. Ann. Probab.. 1997, 25 1514-1531

[6] Haeusler E. On the rate of convergence in the central limit theorem for martingales with discrete and continuous time. Ann. Probab.. 1988, 16 1 275-299

[7] Hall P, Heyde CC. Martingale Limit Theory and Its Applications. 1980 New York: Academic

[8] Heyde CC, Brown BM. On the depature from normality of a certain class of martingales. Ann. Math. Statist.. 1970, 41 2161-2165

[9] Jing BY, Shao QM, Wang Q. Self-normalized Cramér-type large deviations for independent random variables. Ann. Probab.. 2003, 31 4 2167-2215

[10] Lai TL, Wei CZ. Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Ann. Statist.. 1982, 10 1 154-166

[11] Logan BF, Mallows CL, Rice SO, Shepp LA. Limit distributions of self-normalized sums. Ann. Probab.. 1973, 1 788-809

[12] Shao QM. A Cramér type large deviation result for student’s $t-$statistic. J. Theor. Probab.. 1999, 12 2 385-398

[13] Shao QM, Wang QM. Self-normalized limit theorems: a survey. Probab. Surv.. 2013, 10 69-93

[14] Shao QM, Zhou WX. Self-normalization: Taming a wild population in a heavy-tailed world. Appl. Math. J. Chin. Univ.. 2017, 32 253-269

[15] Von 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.. 1965, 36 1 299-303