## Bernoulli

• Bernoulli
• Volume 3, Number 3 (1997), 329-349.

### Berry-Esseen bounds for statistics of weakly dependent samples

#### Abstract

We prove Berry--Esseen bounds for a general class of asymptotically normal statistics which are functions of $N$ weakly dependent random variables under easily verifiable conditions. In particular, we show, for some $δ >0$, the validity of the bound $O (N - 1/2log δ N)$ for $U$-statistics, studentized means, functions of sample means, functionals of empirical distribution functions and linear combinations of order statistics.

#### Article information

Source
Bernoulli, Volume 3, Number 3 (1997), 329-349.

Dates
First available in Project Euclid: 23 April 2007

https://projecteuclid.org/euclid.bj/1177334459

Mathematical Reviews number (MathSciNet)
MR1468309

Zentralblatt MATH identifier
1066.62505

#### Citation

Bentkus, V.; Götze, F.; Tikhomoirov, A. Berry-Esseen bounds for statistics of weakly dependent samples. Bernoulli 3 (1997), no. 3, 329--349. https://projecteuclid.org/euclid.bj/1177334459

#### References

• [1] Bentkus, V. and Götze, F. (1996) The Berry-Esseen bound for Student's statistics. Ann. Probab., 24, 491-503.
• [2] Bentkus, V., Götze, F. and Zitikis, R. (1994) Lower estimates of the convergence rate for U-statistics. Ann. Probab., 22, 1707-1714.
• [3] Bentkus, V., Götze, F. and van Zwet, W. (1997) An Edgeworth expansion for symmetric statistics. Ann. Statist., 25, 851-896.
• [4] Bolthausen, E. and Götze, F. (1993) The rate of convergence for multivariate sampling statistics. Ann. Statist., 21, 1692-1710.
• [5] Cartan, H. (1971) Calcul Differentiel. Formes Differentielles. Paris: Hermann.
• [6] Denker, M. (1982) Statistical decision procedures and ergodic theory. In Ergodic Theory and Related Topics. Mathematical Results, Vitte (GDR), 1981, vol. 12, pp. 35-47. Berlin: Akademie-Verlag.
• [7] Denker, M. and Keller, G. (1983) On U-statistics and von Mises statistics for weakly dependent processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 64, 505-522.
• [8] Eberlein, E. (1984) Weak convergence of partial sums of absolutely regular sequences. Statist. Probab. Lett., 2, 291-293.
• [9] Friedrich, K.O. (1989) A Berry-Esseen bound for functions of independent random variables. Ann. Statist., 17, 170-183.
• [10] Götze, F. (1991) On the rate of convergence in the multivariate CLT. Ann. Probab., 19, 724-739.
• [11] Götze, F. and Hipp, C. (1983) Asymptotic expansions for sums of weakly dependent random vectors. Z. Wahrscheinlichkeitstheorie Verw. Geb., 64, 211-239.
• [12] Götze, F. and Hipp, C. (1994) Asymptotic distribution of statistics in time series. Ann. Statist., 22, 2062-2088.
• [13] Govindarajulu, Z. and Mason, D.M. (1983) A strong representation for linear combinations of order statistics with application to fixed-width confidence intervals for location and scale parameters. Scand. J. Statist., 10, 97-115.
• [14] Heinrich, L. (1992) Bounds for the absolute regularity coefficient of a stationary renewal process. Yokohama Math. J., 40, 25-33.
• [15] Ibragimov, I.A. (1975) A note on the CLT for dependent random variables. Theory Probab. Appl., 20, 135-141.
• [16] Ibragimov, I.A. and Linnik, Yu. V. (1971) Independent and Stationary Sequences of Random Variables. Groningen: Wolters-Noordhoff.
• [17] Rio, E. (1996) Sur le théorème de Berry-Esseen pour les suites faiblement dependantes. Probab. Theory Related Fields, 104, 255-282.
• [18] Stein, C. (1972) A bound on the error in the normal approximation to the distribution of a sum of dependent random variables. In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 2, pp. 583-602. Berkeley: University of California Press.
• [19] Sunklodas, J. (1991) Approximation of distributions of sums of weakly dependent random variables by the normal distribution. In R.V. Gamkrelidze, Yu.V. Prekhorov and V. Statulevicius (eds), Limit Theorems of Probability Theory, Vol. 81, pp. 140-199. Moscow: VINITI.
• [20] Tikhomirov A.N. (1980) On the rule of convergence in the central limit theorem for weakly dependent variables. Theory Probab. Appl., 25, 790-809.
• [21] van Zwet W.R. (1984) A Berry-Esseen bound for symmetric statistics. Z. Wahrscheinlichkeitstheorie. Verw. Geb., 66, 425-440.
• [22] Veretennikov A.Yu. (1987) Bounds for the mixing rate in the theory of stochastic equations. Theory. Probab. Appl., 32, 273-281.
• [23] Yoshihara K.I. (1976) Limit behavior for stationary absolutely regular processes. Z. Wahrscheinlichkeitstheorie Verw. Geb., 35, 237-252.
• [24] Yoshihara K.-I. (1984) The Berry-Esseen theorems for U-statistics generated by absolutely regular processes. Yokohama Math. J., 32, 89-111.