Electronic Communications in Probability

Stein’s method for nonconventional sums

Yeor Hafouta

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Abstract

We obtain almost optimal convergence rate in the central limit theorem for (appropriately normalized) “nonconventional" sums of the form $S_N=\sum _{n=1}^N (F(\xi _n,\xi _{2n},...,\xi _{\ell n})-\bar F)$. Here $\{\xi _n: n\geq 0\}$ is a sufficiently fast mixing vector process with some stationarity conditions, $F$ is bounded Hölder continuous function and $\bar F$ is a certain centralizing constant. Extensions to more general functions $F$ will be discusses, as well. Our approach here is based on the so called Stein’s method, and the rates obtained in this paper significantly improve the rates in [7]. Our results hold true, for instance, when $\xi _n=(T^nf_i)_{i=1}^\wp $ where $T$ is a topologically mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $\{\xi _n: n\geq 0\}$ forms a stationary and exponentially fast $\phi $-mixing sequence, which, for instance, holds true when $\xi _n=(f_i(\Upsilon _n))_{i=1}^\wp $ where $\Upsilon _n$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure.

Article information

Source
Electron. Commun. Probab., Volume 23 (2018), paper no. 38, 14 pp.

Dates
Received: 6 April 2017
Accepted: 5 June 2018
First available in Project Euclid: 9 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.ecp/1528509622

Digital Object Identifier
doi:10.1214/18-ECP142

Mathematical Reviews number (MathSciNet)
MR3820128

Zentralblatt MATH identifier
1397.60061

Subjects
Primary: 60F05: Central limit and other weak theorems

Keywords
central limit theorem Berry–Esseen theorem mixing nonconventional setup Stein’s method

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hafouta, Yeor. Stein’s method for nonconventional sums. Electron. Commun. Probab. 23 (2018), paper no. 38, 14 pp. doi:10.1214/18-ECP142. https://projecteuclid.org/euclid.ecp/1528509622


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References

  • [1] E. Bolthausen, Exact convergence rates in some martingale central limit theorems, Ann. Probab. 10 (1982), 672-688.
  • [2] R.C. Bradley, Introduction to Strong Mixing Conditions, Volume 1, Kendrick Press, Heber City, 2007.
  • [3] L.H.Y Chen and Q.M. Shao, Normal approximation under local dependence, Ann.Probab. 32 (2004), 1985-2028.
  • [4] H. Furstenberg, Nonconventional ergodic averages, Proc. Symp. Pure Math. 50 (1990), 43-56.
  • [5] M.I. Gordin, On the central limit theorem for stationary processes, Soviet Math. Dokl. 10 (1969), 1174-1176.
  • [6] P. Hall, C.C. Heyde, Rates of convergence in the martingale central limit theorem, Ann. Probab. 9 (1981) 395-404.
  • [7] Y. Hafouta and Yu. Kifer, Berry-Esseen type estimates for nonconventional sums, Stoch. Proc. Appl. 126 (2016), 2430-2464.
  • [8] Y. Hafouta and Yu. Kifer, Nonconventional polynomial CLT, Stochastics, 89 (2017), 550-591.
  • [9] Yu. Kifer, Nonconventional limit theorems, Probab. Th. Rel. Fields, 148 (2010), 71-106.
  • [10] Yu. Kifer, Strong approximations for nonconventional sums and almost sure limit theorems, Stochastic Process. Appl., 123 (2013), 2286-2302.
  • [11] Yu. Kifer and S.R.S Varadhan, Nonconventional limit theorems in discrete and continuous time via martingales, Ann. Probab., 42 (2014), 649-688.
  • [12] D.L. McLeish, Invariance principles for dependent variables, Z.Wahrscheinlichkeitstheorie und Verw. Gebiete 32 (1975), 165-178.
  • [13] Y. Rinott, On normal approximation rates for certain sums of dependent random variables, J. Comput. Appl. Math., 55 (1994), 135-143.
  • [14] W. Rudin Real and Complex Analysis, McGraw-Hill, New York, 1987.
  • [15] C. Stein, A bound for the error in the normal approximation to the distribution of a sum of dependent random variables, Proc. Sixth Berkeley Symp. Math. Statist. Probab, 2 (1972), 583-602. Univ. California Press, Berkeley.
  • [16] C. Stein, Approximation Computation of Expectations, IMS, Hayward, CA (1986).
  • [17] N. Shiryaev, Probability, Springer-Verlag, Berlin, 1995.