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Central limit theorems for nearly long range dependent subordinated linear processes

Part of: Limit theorems Inference from stochastic processes Stochastic processes

Published online by Cambridge University Press:  16 July 2020

Martin Wendler  and
Wei Biao Wu
Martin Wendler*
Affiliation:
Otto-von-Guericke-Universität Magdeburg
Wei Biao Wu*
Affiliation:
University of Chicago
*
*Postal address: Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106Magdeburg, Germany. Email: martin.wendler@ovgu.de
**Postal address: University of Chicago, 5747 South Ellis Avenue, Chicago, IL60637, USA
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Abstract

The limit behavior of partial sums for short range dependent stationary sequences (with summable autocovariances) and for long range dependent sequences (with autocovariances summing up to infinity) differs in various aspects. We prove central limit theorems for partial sums of subordinated linear processes of arbitrary power rank which are at the border of short and long range dependence.

Keywords

Central limit theoremlong range dependencelinear processespower rank

MSC classification

Primary: 60F05: Central limit and other weak theorems
Secondary: 60G10: Stationary processes 62M10: Time series, auto-correlation, regression, etc.
Type
Research Papers
Information
Journal of Applied Probability , Volume 57 , Issue 2 , June 2020 , pp. 637 - 656
Copyright
© Applied Probability Trust 2020

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References

Avram, F. andTaqqu, M. S. (1987). Noncentral limit theorems and Appell polynomials. Ann. Prob. 15, 767–775.CrossRefGoogle Scholar
Bai, S. andTaqqu, M. S. (2019). Sensitivity of the Hermite rank. Stochastic Process. Appl. 129, 822–840.CrossRefGoogle Scholar
Billingsley, P. (1968). Convergence of Probability Measures. John Wiley, Chichester.Google Scholar
Bingham, N. H., Goldie, C. M., andTeugels, J. L.(1989). Regular Variation (Encyc. Math. Appl. Vol. 27). Cambridge University Press.Google Scholar
Breuer, P. andMajor, P. (1983). Central limit theorems for non-linear functionals of Gaussian fields. J. Multivariate Anal. 13, 425–441.CrossRefGoogle Scholar
Buchmann, B. andChan, N. H. (2009). Integrated functionals of normal and fractional processes. Ann. Appl. Prob. 19, 49–70.CrossRefGoogle Scholar
Burkholder, D. L. (1966). Martingale transforms. Ann. Math. Statist. 37, 1494–1504.CrossRefGoogle Scholar
Castell, F.M, Guillotin-Plantard, N. andPène, F. (2013). Limit theorems for one and two-dimensional random walks in random scenery. Ann. Inst. H. Poincaré Prob. Statist. 49, 506–528.CrossRefGoogle Scholar
Davydov, Y. A. (1970). The invariance principle for stationary processes. Theory Prob. Appl. 15, 487–498.CrossRefGoogle Scholar
Dobrushin, R. L. andMajor, P. (1979). Non-central limit theorems for non-linear functionals of Gaussian fields. Z. Wahrscheinlichkeitsth. 50, 27–52.CrossRefGoogle Scholar
Ho, H. C. andHsing, T. (1997). Limit theorems for functionals of moving averages. Ann. Prob. 25, 1636–1669.CrossRefGoogle Scholar
Hoeffding, W. andRobbins, H. (1948). The central limit theorem for dependent random variables. Duke Math. J. 15, 773–780.CrossRefGoogle Scholar
Kesten, H. andSpitzer, F. (1979). A limit theorem related to a new class of self similar processes. Z. Wahrscheinlichkeitsth. 50, 5–25.CrossRefGoogle Scholar
Nourdin, Iv., Peccati, G. andReinert, G. (2010). Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos. Ann. Prob. 38, 1947–1985.CrossRefGoogle Scholar
Sly, A. andHeyde, C. (2008). Nonstandard limit theorem for infinite variance functionals. Ann. Prob. 36, 796–805.CrossRefGoogle Scholar
Surgailis, D. (1982). Zones of attraction of self-similar multiple integrals. Lithuanian Math. J. 22, 327–340.CrossRefGoogle Scholar
Taqqu, M. S. (1975). Weak convergence to fractional Brownian motion and to the Rosenblatt process. Z. Wahrscheinlichkeitsth. 31, 287–302.CrossRefGoogle Scholar
Taqqu, M. S. (1979). Convergence of integrated processes of arbitrary Hermite rank. Z. Wahrscheinlichkeitsth. 50, 53–83.CrossRefGoogle Scholar
Wu, W. B. (2003). Empirical processes of long-memory sequences. Bernoulli 9, 809–831.Google Scholar
Wu, W. B., Huang, Y. andZheng, W. (2010). Covariance estimation for long-memory processes. Adv. Appl. Prob. 42, 137–157.CrossRefGoogle Scholar
Wu, W. B. andWoodroofe, W.B. (2004). Martingale approximations for sums of stationary processes. Ann. Prob. 32, 1674–1690.CrossRefGoogle Scholar